Forbidden Subgraphs and Hamiltonian Properties in Graphs
Forbidden Subgraphs and Hamiltonian Properties in Graphs Ron Gould Emory University
Oct. 23, 2009
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
1 0 1 0 1 0 1 0
vertices 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
edges
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
cycle
1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1
deg = 3
1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
degree of a vertex
1 0 1 0
1 0 1 0
deg = 2 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition A cycle that spans the entire vertex set of a graph is called a hamiltonian cycle. Sir William Rowan Hamilton (1865) - the icosian game
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
3 dimensional dodecahedron
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graph of a dodecahedron
00 11 00 11 11111111111111111 00000000000000000 11 00 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00000000000000000 11111111111111111 00 11 000000 111111 00000000000000000 11111111111111111 11 00 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 00 11 00 11 000000 111111 00000000000000000 11111111111111111 00 11 00 11 111111 000000 00000000000000000 11111111111111111 000000 111111 00 11 00 11 00000000000000000 11111111111111111 000000 111111 00 11 00000000000000000 11111111111111111 000000 111111 00 11 11111111111111111 00000000000000000 000000 111111 11 00 000000 111111 000000 111111 00 11 00 11 00 11 000000 111111 00 11 00 11 11 00 000000 00 11 0 1 11 11 11 00 00 00 111111 000000 111111 00 11 0 1 00 11 0 1 0 1 0 1 0 1 00 11 00 11 0 1 00 11 11 00 0 1 000000 111111 11111 00000 00 11 00 11 00 11 00 11 0 1 000000 111111 11111 00000 00 11 00 11 000000 111111 00 11 00 11 000000 111111 000000 111111 000000 111111 00 11 000000 111111 00 11 000000 111111 00 11 00 11 00 11 111111 000000 00 11 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 00 11 000000 111111 00 11 000000 111111 00 11
Ron Gould Emory University
00 11 11 00 00 11
00 11 11 00 00 11
00 11 00 11 00000 11111 00 11 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 11111 00000 00 11 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graph of a dodecahedron
00 11 00 11 11111111111111111 00000000000000000 00000000000000000 11111111111111111 11 00 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 11 00 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00 11 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00 11 111111 000000 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 00000 11111 000 111 000 111 00 11 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000 11111 000 111 000 111 000000 111111 00 11 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000 11111 000 111 000 111 000000 111111 00 11 00000 11111 0000000 1111111 00 11 11111111111111111 00000000000000000 000000 111111 00000 11111 000 111 000 111 000000 111111 00000000 11111111 00 11 00000 11111 0000000 1111111 11 00 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 000000 111111 00000 11111 000 111 000 111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 00 11 00 11 00 11 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 00 11 00 11 00 11 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00 11 0 1 000 111 000111111 111 00000 11111 0000000 1111111 11 11 11 00 00 00 000000 000000 111111 00 11 000 111 000 111 0 1 00000 11111 1111111 0 10000000 000000 1 111111 00 11111 11 0 000 111 000 111 00000 0 1 000000 111111 0 000 111 000 111 00000 11111 0 1 000000 1 111111 0 1 000 111 000 111 00000 11111 0 1 000000 111111 0 000 111 000 111 00000 11111 0 1 000000 1 111111 00 11 00 11 0 1 000 111 000 111 00000 11111 0 1 000000 111111 00 11 00 11 000000 1 000 111 000 111 00000 11111 0 1 11111 000000 111111 000000 111111 11111 00000 000000 11 111111 00 00 11 00 11 00 11 000000 1 00000 11111 0 1 11111 000000 111111 000000 111111 11111 00000 000000 111111 00 11 00 11 00000 11111 000 111111 111 00 11 000000 000000 111111 000000 11 111111 00 00 11 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111111 111 00 11 000000 000000 111111 000000 111111 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111111 111 00 11 00 11 000000 000000 111111 000000 111111 00000 11111 000 111 00 11 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111 00 11 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00 11 00 11 00 11 00000 11111 000 111 00 11 000000 111111 00 11 00 11 00 11 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00 00 00 11 11 11 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 00 11 00 11 000000 111111 000000 111111 000000 111111 11111 00000 00 11 00 11 000000 111111 000000 111111 00 11 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sir William Rowan Hamilton
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Early results
Theorem Dirac, 1952 If G is a graph of order n ≥ 3 with δ(G ) ≥ n/2 then G is hamiltonian. Theorem Ore, 1960 If G is a graph of order n ≥ 3 with degx + degy ≥ n for every pair of nonadjacent vertices x, y , then G is hamiltonian.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced subgraphs
Definition A subgraph H of a graph G is induced if H is a subgraph of G and e is an edge of H if and only if e is an edge of G .
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
1 0 1 0
1 0 1 0 1 0 0 1 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
00000 11111 00000 11111 1 0 1 0 00000 11111 00000 11111 1 1 0 0 00000 11111 00000 0000011111 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 000000000 111111111 0 1 00000 11111 00000 11111 000000000 111111111 0 1 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 00 11 0000000 1111111 0 1 000000000 111111111 00 11 0000000 1111111 0 1 0000000 1111111 00 11 0000000 1111111 00 000000011 1111111 00 11 0000000 1111111 00 11 00 11 0000000 1111111 000000011 1111111 00 00 0000000 1111111 000000011 1111111 00 11 000000011 1111111 00 0 1 0 1 11 00 0 1 0 1
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Basics
Definition A graph G of order n is pancyclic if G contains cycles of all possible lengths from 3 to n. Definition We say the set H = {H1 , . . . , Hk } is forbidden in G if no Hi is an induced subgraph of G . We also say G is H-free.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs of interest Graphs of interest here include:
claw K 1,3 net N
Ron Gould Emory University
1,1,1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs of interest Graphs of interest include:
P
6
Ron Gould Emory University
N(3,2,1)
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Very Ancient History
Goodman and Hedetniemi, 1974 Theorem If G is a 2-connected { K1,3 , N(1, 0, 0) }-free graph, then G is hamiltonian.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Ancient History -
Since the Goodman - Hedetniemi result in 1974, a great many results have occurred concerning families of forbidden subgraphs implying various cycle (hamiltonian) properties. The first such result was the following: Theorem D. Duffus, RG, M. Jacobson, 1980. If G is a {K1,3 , N(1, 1, 1)}-free graph, then (a) if G is 2-connected, then G is hamiltonian; (b) if G is connected, then G is traceable.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 00000000 11111111 0 1 0 1 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 11 1 0 1 0 1 0 00 00 1 00 11 00 11 00 00 11 00 0 1 0 11 1 0 11 1 00 11 11 00 00 00 11 00 11 11 00 11 11 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11 00 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 0 1 000000 111111 00000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 11 1 000000 111111 0 1 00000 11111 00000 11111 0 1 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 11 00 00 00 00 11 11 00 11 11 00 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11 00 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 0 1 000000 111111 00000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 11 1 000000 111111 0 1 00000 11111 00000 11111 0 1 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 11 00 00 00 00 11 11 00 11 11 00 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 11 1 000000 111111 0 1 0 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 00000 11111 00000 11111 00000 11111 00 11 00000 00000 11111 11111
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11 00 00 11
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 00 11 00 00 0 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11 00 00 11
11111111 00000000 00000000 11111111 00000000 11111111 00 00000000 11111111 00000000 11111111 0 11 1 00 11 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 00 11 00 00 0 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
00000000 11111111 11111111 00000000 11111111 00000000 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 000000 111111 0 11111111 1 0 1 000000 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 111111 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 0 1 000000 0000 1111 000000 111111 0 111111 1 00000 1 11111 0 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 11111 1 000000 111111 0000 1111 000000 111111 0 111111 1 00000 11111 0 1 000000 111111 00000 11111 0 1 0 1 00000 11111 00000 000000 0000 1111 000000 111111 0 1 11 00 00 11 00 11 00 11 00 11 00 11 00 00 11 00 00000 11111 0 00 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00000 11111 0000 1111 11 00 11 11 11 00 11 00 11 11 00000 1 11111 0 1 0 11111 1 0000000 11111 00000 11111 0000 1111 00000 11111 0 0 1 00000 11111 00000 0000 1111 0 1 0 11111 1 00000 00000 11111 00000 11111 0000 1111 00000 1 11111 0 0 1 00000 11111 00000 11111 0000 1111 0 1 0 11111 1 00000 11111 00000 11111 0000 1111 0 1 0 1 00000 00000 11111 0000 1111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 00000 11111 11111 00000 11111
Ron Gould Emory University
00000000 11111111 11111111 00000000 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 11 00 11 00 11 00 0 1 0 1 0 1 0 1 11 00 11 00 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
00000000 11111111 11111111 00000000 11111111 00000000 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 000000 111111 0 11111111 1 0 1 000000 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 111111 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 0 1 000000 0000 1111 000000 111111 0 111111 1 00000 1 11111 0 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 11111 1 000000 111111 0000 1111 000000 111111 0 111111 1 00000 11111 0 1 000000 111111 00000 11111 0 1 0 1 00000 11111 00000 000000 0000 1111 000000 111111 0 1 11 00 00 11 00 11 00 11 00 11 00 11 00 00 11 00 00000 11111 0 00 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00000 11111 0000 1111 11 00 11 11 11 00 11 00 11 11 00000 1 11111 0 1 0 11111 1 0000000 11111 00000 11111 0000 1111 00000 11111 0 0 1 00000 11111 00000 0000 1111 0 1 0 11111 1 00000 00000 11111 00000 11111 0000 1111 00000 1 11111 0 0 1 00000 11111 00000 11111 0000 1111 0 1 0 11111 1 00000 11111 00000 11111 0000 1111 0 1 0 1 00000 00000 11111 0000 1111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 00000 11111 11111 00000 11111
Ron Gould Emory University
00000000 11111111 11111111 00000000 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 11 00 11 00 11 00 0 1 0 1 0 1 0 1 11 00 11 00 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6 (Bedrosian, 1991) K1,3 , N(2, 1, 0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6 (Bedrosian, 1991) K1,3 , N(2, 1, 0) (Faudree, RG, Ryj´ a˘ cek, Schiermeyer, 1995) K1,3 , N(3, 0, 0) for n ≥ 10.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Hamiltonian Connected Extension
Definition A graph G is called hamiltonian connected if any pair of distinct vertices are joined by a spanning path. Theorem (F.B. Shepard, 1991) If G is a 3-connected claw and net free graph, then G is hamiltonian connected.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Nets N(i, j, k) with i + j + k = 3
Theorem Bedrosian 91 - R. Faudree and RG 97. Let R and S be connected graphs (R, S 6= P3 ) and G a 2-connected graph of order n. Then G is {R, S}-free implies G is hamiltonian if, and only if, R = K1,3 and S is one of the graphs N(1, 1, 1), P6 , N(2, 1, 0), N(2, 0, 0), (or N(3, 0, 0) when n ≥ 10), or a connected induced subgraph of one of these graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
No claw and no Z(3,0,0), but not hamiltonian
P
Ron Gould Emory University
T,T,T
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Z
2
= N(2,0,0) W = N(2,1,0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Question: What changes if we consider 3-connected graphs? We also turn our attention to pancyclic graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
A new graph is now needed.
L
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Modern times call for bigger nets!
Theorem RG, T. Luczak, F. Pfender, 2004. Let X and Y be connected graphs on at least three vertices such that X , Y 6= P3 and Y 6= K1,3 . Then the following statements are equivalent: 1
Every 3-connected {X , Y }-free graph G is pancyclic.
2
X = K1,3 and Y is a subgraph of one of the graphs from the family F = {P7 , L, N(4, 0, 0), N(3, 1, 0), N(2, 2, 0), N(2, 1, 1)}.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
The proof
Comment 1: There are two phases to proving this result.
Phase 1: Show that for each graph Y from F, each 3-connected {K1,3 , Y }-free graph is pancyclic.
Phase 2: Show that no other pair works!
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
G0
K3,3
G2 G1
G3
3-connected non-pancyclic graphs Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition A graph G is subpancyclic if for every k, 3 ≤ k ≤ c(G ) G contains a cycle of length k.
Here c(G ) is the circumference of G , that is, the length of a longest cycle.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition For k-connected graphs, let Fk be the family of all graphs G such that each claw-free k-connected graph, which contains no induced copy of G is subpancyclic, except possibly a finite number of exceptions (sufficiently large).
Note: F 1 = F2 .
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
In order to find F2 note the following: Cn≥4 is 2-connected but not subpancyclic, so the only possible members of F2 are paths. Consider the graph Gn composed of K2n and a perfect matching on 2n vertices, and an additional perfect matching between these two graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
The graph below is hamiltonian, but contains no cycles on 4n − 1 vertices, and no induced P7 .
K 2n
...
nK 2
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Finally, it is well known that each 2-connected {K1,3 , P6 }−free graph n ≥ 10 vertices is pancyclic and hence subpancyclic.
We conclude that F2 = {P3 , P4 , P5 , P6 }
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
We studied Fk for k ≥ 3. To describle Fk we need two more graphs: Let L(r , s) denote the graph consisting of two disjoint cliques on r vertices connected by a path of length s (generalizing L).
Kr s
L(r,s) Kr
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
By F (t1 , t2 , . . . , tr ) with 0 ≤ t1 ≤ . . . ≤ tr we mean a complete graph on r vertices with paths of length t1 , t2 , . . . , tr rooted at different vertices of the Kr (generalizing N(i, j, k)).
Kr
...
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Theorem RG, Luczak, Pfender. Let k ≥ 3 and r = r (k) = ⌈k/2 + 1⌉. Then there exists m = m(k) such that every G ∈ Fk is a subgraph of either L(r , 2s + 1) for some s ≥ 0 or F (t1 , . . . , tr ) with 0 ≤ t1 ≤ . . . ≤ tr −2 ≤ m
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
We also study in more detail the family F3 . Note: we already know some graphs that must belong, namely P7 , L(3, 1) = L, F(0, 0, 4) = N(4, 0, 0), F(0, 1, 3), F(1, 1, 2) We show that: Theorem RG, T. Luczak, F. Pfender. The family F3 contains all the graphs F (t1 , t2 , t3 ) with t1 ≤ 2 as well as the graphs L(3, 2s + 1) for s ≥ 0.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
One final Question: What about 4-connected? Now we will consider the pairs that worked for 3-connected graphs and extend that family.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
with M. Ferrara, S. Gehrke, C. Magnant, J. Powell.
Now we consider the generalized nets N(i, j, k) where i + j + k = 5. Theorem Let G be a 4-connected {K1,3 , N(i, j, k)}-free graph with i + j + k = 5. Then G is pancyclic.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Clearly, we had to show that such graphs are hamiltonian. This builds on the set of forbidden pairs that imply a 4-connected graph is hamiltonian. Conjecture (Matthews and Sumner, 1985) Every 4-connected claw-free graph is hamiltonian. Question 1. Can we characterize the 4-connected pairs that imply a graph is pancyclic? 2. Are there pairs not involving the claw?
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Oct. 23, 2009
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
1 0 1 0 1 0 1 0
vertices 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
edges
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
cycle
1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs - Some Basics 1 0 0 1 1 0 0 1
1 0 0 1
deg = 3
1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1
1 0 1 0
1 0 1 0
degree of a vertex
1 0 1 0
1 0 1 0
deg = 2 1 0 1 0 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition A cycle that spans the entire vertex set of a graph is called a hamiltonian cycle. Sir William Rowan Hamilton (1865) - the icosian game
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
3 dimensional dodecahedron
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graph of a dodecahedron
00 11 00 11 11111111111111111 00000000000000000 11 00 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00000000000000000 11111111111111111 00 11 000000 111111 00000000000000000 11111111111111111 11 00 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 000000 111111 00000000000000000 11111111111111111 00 11 00 11 000000 111111 00000000000000000 11111111111111111 00 11 00 11 111111 000000 00000000000000000 11111111111111111 000000 111111 00 11 00 11 00000000000000000 11111111111111111 000000 111111 00 11 00000000000000000 11111111111111111 000000 111111 00 11 11111111111111111 00000000000000000 000000 111111 11 00 000000 111111 000000 111111 00 11 00 11 00 11 000000 111111 00 11 00 11 11 00 000000 00 11 0 1 11 11 11 00 00 00 111111 000000 111111 00 11 0 1 00 11 0 1 0 1 0 1 0 1 00 11 00 11 0 1 00 11 11 00 0 1 000000 111111 11111 00000 00 11 00 11 00 11 00 11 0 1 000000 111111 11111 00000 00 11 00 11 000000 111111 00 11 00 11 000000 111111 000000 111111 000000 111111 00 11 000000 111111 00 11 000000 111111 00 11 00 11 00 11 111111 000000 00 11 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 00 11 000000 111111 00 11 000000 111111 00 11
Ron Gould Emory University
00 11 11 00 00 11
00 11 11 00 00 11
00 11 00 11 00000 11111 00 11 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 11111 00000 00 11 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graph of a dodecahedron
00 11 00 11 11111111111111111 00000000000000000 00000000000000000 11111111111111111 11 00 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 0000000000000000011111111111111111 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 11 00 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00 11 000000 111111 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00 11 00 11 111111 000000 000000 111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 000000 111111 00000 11111 000 111 000 111 00 11 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000 11111 000 111 000 111 000000 111111 00 11 00 11 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000 11111 000 111 000 111 000000 111111 00 11 00000 11111 0000000 1111111 00 11 11111111111111111 00000000000000000 000000 111111 00000 11111 000 111 000 111 000000 111111 00000000 11111111 00 11 00000 11111 0000000 1111111 11 00 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 000000 111111 00000 11111 000 111 000 111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 00 11 00 11 00 11 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00000 11111 0000000 1111111 00 11 00 11 00 11 00000 11111 000 111 000 111 000000 111111 000000 111111 00000000 11111111 00 11 0 1 000 111 000111111 111 00000 11111 0000000 1111111 11 11 11 00 00 00 000000 000000 111111 00 11 000 111 000 111 0 1 00000 11111 1111111 0 10000000 000000 1 111111 00 11111 11 0 000 111 000 111 00000 0 1 000000 111111 0 000 111 000 111 00000 11111 0 1 000000 1 111111 0 1 000 111 000 111 00000 11111 0 1 000000 111111 0 000 111 000 111 00000 11111 0 1 000000 1 111111 00 11 00 11 0 1 000 111 000 111 00000 11111 0 1 000000 111111 00 11 00 11 000000 1 000 111 000 111 00000 11111 0 1 11111 000000 111111 000000 111111 11111 00000 000000 11 111111 00 00 11 00 11 00 11 000000 1 00000 11111 0 1 11111 000000 111111 000000 111111 11111 00000 000000 111111 00 11 00 11 00000 11111 000 111111 111 00 11 000000 000000 111111 000000 11 111111 00 00 11 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111111 111 00 11 000000 000000 111111 000000 111111 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111111 111 00 11 00 11 000000 000000 111111 000000 111111 00000 11111 000 111 00 11 00 11 000000 111111 000000 111111 000000 111111 00000 11111 000 111 00 11 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00000 11111 000 111 00 11 000000 111111 00 11 00 11 00 11 00000 11111 000 111 00 11 000000 111111 00 11 00 11 00 11 00000 11111 000 111 00 11 000000 111111 000000 111111 000000 111111 00 00 00 11 11 11 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 00 11 00 11 000000 111111 000000 111111 000000 111111 11111 00000 00 11 00 11 000000 111111 000000 111111 00 11 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sir William Rowan Hamilton
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Early results
Theorem Dirac, 1952 If G is a graph of order n ≥ 3 with δ(G ) ≥ n/2 then G is hamiltonian. Theorem Ore, 1960 If G is a graph of order n ≥ 3 with degx + degy ≥ n for every pair of nonadjacent vertices x, y , then G is hamiltonian.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced subgraphs
Definition A subgraph H of a graph G is induced if H is a subgraph of G and e is an edge of H if and only if e is an edge of G .
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
1 0 1 0
1 0 1 0 1 0 0 1 1 0 0 1
1 0 0 1
Ron Gould Emory University
1 0 0 1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Induced Subgraphs - Examples
00000 11111 00000 11111 1 0 1 0 00000 11111 00000 11111 1 1 0 0 00000 11111 00000 0000011111 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 000000000 111111111 0 1 00000 11111 00000 11111 000000000 111111111 0 1 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 00 11 0000000 1111111 0 1 000000000 111111111 00 11 0000000 1111111 0 1 0000000 1111111 00 11 0000000 1111111 00 000000011 1111111 00 11 0000000 1111111 00 11 00 11 0000000 1111111 000000011 1111111 00 00 0000000 1111111 000000011 1111111 00 11 000000011 1111111 00 0 1 0 1 11 00 0 1 0 1
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Basics
Definition A graph G of order n is pancyclic if G contains cycles of all possible lengths from 3 to n. Definition We say the set H = {H1 , . . . , Hk } is forbidden in G if no Hi is an induced subgraph of G . We also say G is H-free.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs of interest Graphs of interest here include:
claw K 1,3 net N
Ron Gould Emory University
1,1,1
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Graphs of interest Graphs of interest include:
P
6
Ron Gould Emory University
N(3,2,1)
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Very Ancient History
Goodman and Hedetniemi, 1974 Theorem If G is a 2-connected { K1,3 , N(1, 0, 0) }-free graph, then G is hamiltonian.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Proof y− 11 00 00 11 00 11 11 00 00 11 00 11
y 00 11 11 00 00 11 00 11 11 00 00 11
y+
11 00 00 11 00 11 11 00 00 11 00 11
x 11 00 00 11 00 11
00 11 11 00 00 11 00 11 11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Ancient History -
Since the Goodman - Hedetniemi result in 1974, a great many results have occurred concerning families of forbidden subgraphs implying various cycle (hamiltonian) properties. The first such result was the following: Theorem D. Duffus, RG, M. Jacobson, 1980. If G is a {K1,3 , N(1, 1, 1)}-free graph, then (a) if G is 2-connected, then G is hamiltonian; (b) if G is connected, then G is traceable.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 00000000 11111111 0 1 0 1 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 11 1 0 1 0 1 0 00 00 1 00 11 00 11 00 00 11 00 0 1 0 11 1 0 11 1 00 11 11 00 00 00 11 00 11 11 00 11 11 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11 00 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 0 1 000000 111111 00000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 11 1 000000 111111 0 1 00000 11111 00000 11111 0 1 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 11 00 00 00 00 11 11 00 11 11 00 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11 00 00 11
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 0 1 000000 111111 00000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 11 1 000000 111111 0 1 00000 11111 00000 11111 0 1 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 11 00 00 00 00 11 11 00 11 11 00 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 0 111111 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 1 0 1 000000 0 1 000000 000000 111111 0 111111 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 11 1 000000 111111 0 1 0 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 11 00 11 00 00 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 00000 11111 00000 11111 00000 11111 00 11 00000 00000 11111 11111
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
11 00 00 11
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11 00 00 11
11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 00 11 00 00 0 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
11 00 00 11
11111111 00000000 00000000 11111111 00000000 11111111 00 00000000 11111111 00000000 11111111 0 11 1 00 11 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 111111 1 000000 111111 000000 111111 0 111111 1 0 1 000000 0 1 000000 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 11 1 0 11 1 00000 11111 00000 11111 000000 111111 000000 111111 0 1 00 11 00 11 00 00 00 11 00 00 0 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00 11 00 00 00 00 11 00 11 00 11 11 11 11 11 11 0 1 0 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 0 1 0 1 00000 11111 00000 0 1 0 1 00000 11111 00000 00000 11111 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000
Ron Gould Emory University
11111111 00000000 00000000 11111111 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 0 1 0 1 00 11 00 11 00 11 0 1 0 1 0 1 0 1 00 11 00 00 11 11 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
00000000 11111111 11111111 00000000 11111111 00000000 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 000000 111111 0 11111111 1 0 1 000000 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 111111 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 0 1 000000 0000 1111 000000 111111 0 111111 1 00000 1 11111 0 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 11111 1 000000 111111 0000 1111 000000 111111 0 111111 1 00000 11111 0 1 000000 111111 00000 11111 0 1 0 1 00000 11111 00000 000000 0000 1111 000000 111111 0 1 11 00 00 11 00 11 00 11 00 11 00 11 00 00 11 00 00000 11111 0 00 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00000 11111 0000 1111 11 00 11 11 11 00 11 00 11 11 00000 1 11111 0 1 0 11111 1 0000000 11111 00000 11111 0000 1111 00000 11111 0 0 1 00000 11111 00000 0000 1111 0 1 0 11111 1 00000 00000 11111 00000 11111 0000 1111 00000 1 11111 0 0 1 00000 11111 00000 11111 0000 1111 0 1 0 11111 1 00000 11111 00000 11111 0000 1111 0 1 0 1 00000 00000 11111 0000 1111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 00000 11111 11111 00000 11111
Ron Gould Emory University
00000000 11111111 11111111 00000000 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 11 00 11 00 11 00 0 1 0 1 0 1 0 1 11 00 11 00 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Sketch of proof.
00000000 11111111 11111111 00000000 11111111 00000000 00000000 11111111 00000000 11111111 0 1 00000000 11111111 000000 111111 0 11111111 1 00000000 11111111 0 1 000000 111111 00000000 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 1 000000 111111 00000000 11111111 000000 111111 0 1 00000000 11111111 0 1 000000 111111 0 111111 1 000000 111111 00000000 000000 111111 0 11111111 1 0 1 000000 0 1 000000 111111 000000 111111 0 1 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 111111 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 0 1 000000 0000 1111 000000 111111 0 111111 1 00000 1 11111 0 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 1 000000 111111 0000 1111 000000 111111 0 1 00000 11111 0 1 000000 111111 0 11111 1 000000 111111 0000 1111 000000 111111 0 111111 1 00000 11111 0 1 000000 111111 00000 11111 0 1 0 1 00000 11111 00000 000000 0000 1111 000000 111111 0 1 11 00 00 11 00 11 00 11 00 11 00 11 00 00 11 00 00000 11111 0 00 1 000000 111111 0 1 0 1 00000 11111 00000 11111 00000 11111 0000 1111 11 00 11 11 11 00 11 00 11 11 00000 1 11111 0 1 0 11111 1 0000000 11111 00000 11111 0000 1111 00000 11111 0 0 1 00000 11111 00000 0000 1111 0 1 0 11111 1 00000 00000 11111 00000 11111 0000 1111 00000 1 11111 0 0 1 00000 11111 00000 11111 0000 1111 0 1 0 11111 1 00000 11111 00000 11111 0000 1111 0 1 0 1 00000 00000 11111 0000 1111 0 1 0 1 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 0 1 0 11111 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 00000 11111 11111 00000 11111
Ron Gould Emory University
00000000 11111111 11111111 00000000 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 00000000 11111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 00 11 00 11 00 11 00 0 1 0 1 0 1 0 1 11 00 11 00 11 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 0 1 00000 11111 00000 11111 00000 11111 00000 11111
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6 (Bedrosian, 1991) K1,3 , N(2, 1, 0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
This Theorem spurred interest in the area and a number of other results soon followed. Of special interest are the following pairs, each of which implies a 2-connected G is hamiltonian: (RG, Jacobson, 1982) K1,3 , N(2, 0, 0) (Broersma and Veldman, 1990) K1,3 , P6 (Bedrosian, 1991) K1,3 , N(2, 1, 0) (Faudree, RG, Ryj´ a˘ cek, Schiermeyer, 1995) K1,3 , N(3, 0, 0) for n ≥ 10.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Hamiltonian Connected Extension
Definition A graph G is called hamiltonian connected if any pair of distinct vertices are joined by a spanning path. Theorem (F.B. Shepard, 1991) If G is a 3-connected claw and net free graph, then G is hamiltonian connected.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Nets N(i, j, k) with i + j + k = 3
Theorem Bedrosian 91 - R. Faudree and RG 97. Let R and S be connected graphs (R, S 6= P3 ) and G a 2-connected graph of order n. Then G is {R, S}-free implies G is hamiltonian if, and only if, R = K1,3 and S is one of the graphs N(1, 1, 1), P6 , N(2, 1, 0), N(2, 0, 0), (or N(3, 0, 0) when n ≥ 10), or a connected induced subgraph of one of these graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
No claw and no Z(3,0,0), but not hamiltonian
P
Ron Gould Emory University
T,T,T
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Z
2
= N(2,0,0) W = N(2,1,0)
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Question: What changes if we consider 3-connected graphs? We also turn our attention to pancyclic graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
A new graph is now needed.
L
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Modern times call for bigger nets!
Theorem RG, T. Luczak, F. Pfender, 2004. Let X and Y be connected graphs on at least three vertices such that X , Y 6= P3 and Y 6= K1,3 . Then the following statements are equivalent: 1
Every 3-connected {X , Y }-free graph G is pancyclic.
2
X = K1,3 and Y is a subgraph of one of the graphs from the family F = {P7 , L, N(4, 0, 0), N(3, 1, 0), N(2, 2, 0), N(2, 1, 1)}.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
The proof
Comment 1: There are two phases to proving this result.
Phase 1: Show that for each graph Y from F, each 3-connected {K1,3 , Y }-free graph is pancyclic.
Phase 2: Show that no other pair works!
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
G0
K3,3
G2 G1
G3
3-connected non-pancyclic graphs Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition A graph G is subpancyclic if for every k, 3 ≤ k ≤ c(G ) G contains a cycle of length k.
Here c(G ) is the circumference of G , that is, the length of a longest cycle.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Definition For k-connected graphs, let Fk be the family of all graphs G such that each claw-free k-connected graph, which contains no induced copy of G is subpancyclic, except possibly a finite number of exceptions (sufficiently large).
Note: F 1 = F2 .
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
In order to find F2 note the following: Cn≥4 is 2-connected but not subpancyclic, so the only possible members of F2 are paths. Consider the graph Gn composed of K2n and a perfect matching on 2n vertices, and an additional perfect matching between these two graphs.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
The graph below is hamiltonian, but contains no cycles on 4n − 1 vertices, and no induced P7 .
K 2n
...
nK 2
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Finally, it is well known that each 2-connected {K1,3 , P6 }−free graph n ≥ 10 vertices is pancyclic and hence subpancyclic.
We conclude that F2 = {P3 , P4 , P5 , P6 }
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
We studied Fk for k ≥ 3. To describle Fk we need two more graphs: Let L(r , s) denote the graph consisting of two disjoint cliques on r vertices connected by a path of length s (generalizing L).
Kr s
L(r,s) Kr
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
By F (t1 , t2 , . . . , tr ) with 0 ≤ t1 ≤ . . . ≤ tr we mean a complete graph on r vertices with paths of length t1 , t2 , . . . , tr rooted at different vertices of the Kr (generalizing N(i, j, k)).
Kr
...
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Theorem RG, Luczak, Pfender. Let k ≥ 3 and r = r (k) = ⌈k/2 + 1⌉. Then there exists m = m(k) such that every G ∈ Fk is a subgraph of either L(r , 2s + 1) for some s ≥ 0 or F (t1 , . . . , tr ) with 0 ≤ t1 ≤ . . . ≤ tr −2 ≤ m
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
We also study in more detail the family F3 . Note: we already know some graphs that must belong, namely P7 , L(3, 1) = L, F(0, 0, 4) = N(4, 0, 0), F(0, 1, 3), F(1, 1, 2) We show that: Theorem RG, T. Luczak, F. Pfender. The family F3 contains all the graphs F (t1 , t2 , t3 ) with t1 ≤ 2 as well as the graphs L(3, 2s + 1) for s ≥ 0.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
One final Question: What about 4-connected? Now we will consider the pairs that worked for 3-connected graphs and extend that family.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
with M. Ferrara, S. Gehrke, C. Magnant, J. Powell.
Now we consider the generalized nets N(i, j, k) where i + j + k = 5. Theorem Let G be a 4-connected {K1,3 , N(i, j, k)}-free graph with i + j + k = 5. Then G is pancyclic.
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
Clearly, we had to show that such graphs are hamiltonian. This builds on the set of forbidden pairs that imply a 4-connected graph is hamiltonian. Conjecture (Matthews and Sumner, 1985) Every 4-connected claw-free graph is hamiltonian. Question 1. Can we characterize the 4-connected pairs that imply a graph is pancyclic? 2. Are there pairs not involving the claw?
Ron Gould Emory University
Forbidden Subgraphs and Hamiltonian Properties in Graphs
