Minimum Degree and Disjoint Cycles in Claw-free Graphs
Minimum Degree and Disjoint Cycles in Claw-free Graphs Ron Gould Emory University
April 30, 2011
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about :
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
◮
hamiltonicity,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
◮
hamiltonicity,
◮
general cycle structure, and more.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Commonly Forbidden Graphs
Claw K
1,3
P6
The net N(1,1,1) Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Here we are interested in vertex disjoint cycles. In particular, we are interested in questions about the maximum number of disjoint triangles we can obtain in various settings, as well as when we can cover G with disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem H. Wang, 1998 For any integer k ≥ 2, if G is a claw-free graph of order n ≥ 6(k − 1) with δ(G ) ≥ 3, then G contains at least k vertex disjoint triangles or belongs to a special family that has only k − 1 disjoint triangles.
So about 1/2 of the possible disjoint triangles are there.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem G. Chen, L. Markus and R. Schelp, 1995 Let k ≥ 1 and G be a K1,r -free graph of order n and size q. 1. If r = 3 and q ≥ n + 12 (3k − 1)(3k − 4) + 1, then G contains k vertex disjoint cycles. 2. If r ≥ 4 and q ≥ n + 16rk 2 , then G contains k vertex disjoint cycles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem G. Chen, J. Faudree, RG, A. Saito, 2000 If G is a 2-connected claw-free graph with δ(G ) ≥
n−2 , 3
then G contains a 2-factor with exactly k cycles for 1 ≤ k ≤ n−24 3 . Furthermore, this result is sharp in the sense that if we lower δ(G ) we cannot obtain the full range of values for k.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
What next?
Question What about disjoint triangles under higher minimum degree conditions? Wang consider the smallest degree condition, we wish to be more general.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of order n and minimum degree δ, then G contains at least (
δ−2 n ) δ+1 3
vertex disjoint triangles.
Note for small δ Wang’s result is better but at δ = 5 we essentially achieve Wang’s bound and exceed it after that.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Special Case!
There is a special case of the last result of interest.
Corollary If G is a claw-free graph of order n with minimum degree δ(G ) ≥ n/3, then G contains at least n/3 − 2 vertex disjoint triangles and this result is best possible.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Sharpness Let n = 9k + 6 so that n/3 = 3k + 2 and let |A| = |B| = |C | = n/3.
.. .
.. .
.. .
A
B
C
Figure: Graph with δ = n/3 and n/3 − 2 triangles. Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Question What δ is required for a 2-factor with exactly two cycles?
... K(n−1)/3
...
...
K
(n−4)/3
... K
(n−1)/3
Figure: G with no 2-factor of two cycles, δ =
Ron Gould Emory University
n−1 3 .
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Theorem R.J. Faudree, RG, M.S. Jacobson Let k be a positive integer. If G is a claw-free graph of order n ≥ 2k 4 − 2k 2 + k with δ(G ) ≥ n/k, then G contains a 2-factor with k − 1 components. Further, this value of δ(G ) is best possible.
The case k = 3 say δ ≥ n/3 impies a 2-factor with exactly 2 cycles, which is best possible by the earlier example.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Can we say more about 2-factors? Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of sufficiently large order n with δ(G ) ≥ n/3, then G contains a 2-factor with k disjoint cycles, for 2 ≤ k ≤ ⌊n/3 − 2⌋.
Note: no connectivity condition here! Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Recall a classic result.
Theorem Corradi and Hajnal, 1963 If G is a graph of order n = 3k with δ(G ) ≥ 2n/3, then G contains k disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
The claw-free case, here n = 3k. K 3t+2
... ... ... K 3t+1 Figure: No triangle cover and δ < n/2.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
A claw-free C-H type theorem
Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of sufficiently large order n = 3k with δ(G ) ≥ n/2, then G contains k disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Sketch of proof
Theorem Li, Rousseau and Zang, 2000 The ramsey number r (Kk , Kn ) ≤ (1 + o(1))
Ron Gould Emory University
nk−1 . (log n)k−2
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Useful Lemma
Lemma In a claw-free graph G with δ(G ) ≥ n/k, and k ≥ 2, then α(G ) ≤ 2k − 1.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
m≈
√
n−ǫ
B2 deg ≥ 17 to C Km
B1 ≥ n/2 − o(n)
remaining vertices i.e.
C = G − (A ∪ B) |C| ≥ n/2 − o(n)
A
B degree at most 16 to A
Ron Gould Emory University
C
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Steps in proof:
1. Find one or two vertices in A ∪ B2 ∪ C with three or more adjacencies into B1 . Since claw-free then we get a triangle using 2 vertices of B1 . 2. We reduce B1 until multiple of 3, then by denseness of it, we apply C-H to get triangle cover of that part.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Steps in proof:
3. Now as r (K3 , K4 ) = 9, we can place vertices of B2 in triangles until at most 8 remain. Each of these has degree at least 17 into C and can be placed on triangles using C . Now do a similar thing with C (using r = 9) until at most 8 remain. But these 8 have degree at least 17 in A and can be placed on triangles. 4. Finally, what remains of A is a clique and has order a multiple of 3, so we may easily finish.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
April 30, 2011
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about :
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
◮
hamiltonicity,
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Definition We say a graph G is H-free if G contains no induced copy of H as a subgraph. Using forbidden subgraphs, people have studied questions about : ◮
chromatic number,
◮
matchings,
◮
paths,
◮
hamiltonicity,
◮
general cycle structure, and more.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Commonly Forbidden Graphs
Claw K
1,3
P6
The net N(1,1,1) Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Here we are interested in vertex disjoint cycles. In particular, we are interested in questions about the maximum number of disjoint triangles we can obtain in various settings, as well as when we can cover G with disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem H. Wang, 1998 For any integer k ≥ 2, if G is a claw-free graph of order n ≥ 6(k − 1) with δ(G ) ≥ 3, then G contains at least k vertex disjoint triangles or belongs to a special family that has only k − 1 disjoint triangles.
So about 1/2 of the possible disjoint triangles are there.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem G. Chen, L. Markus and R. Schelp, 1995 Let k ≥ 1 and G be a K1,r -free graph of order n and size q. 1. If r = 3 and q ≥ n + 12 (3k − 1)(3k − 4) + 1, then G contains k vertex disjoint cycles. 2. If r ≥ 4 and q ≥ n + 16rk 2 , then G contains k vertex disjoint cycles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Some old results of interest
Theorem G. Chen, J. Faudree, RG, A. Saito, 2000 If G is a 2-connected claw-free graph with δ(G ) ≥
n−2 , 3
then G contains a 2-factor with exactly k cycles for 1 ≤ k ≤ n−24 3 . Furthermore, this result is sharp in the sense that if we lower δ(G ) we cannot obtain the full range of values for k.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
What next?
Question What about disjoint triangles under higher minimum degree conditions? Wang consider the smallest degree condition, we wish to be more general.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of order n and minimum degree δ, then G contains at least (
δ−2 n ) δ+1 3
vertex disjoint triangles.
Note for small δ Wang’s result is better but at δ = 5 we essentially achieve Wang’s bound and exceed it after that.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Special Case!
There is a special case of the last result of interest.
Corollary If G is a claw-free graph of order n with minimum degree δ(G ) ≥ n/3, then G contains at least n/3 − 2 vertex disjoint triangles and this result is best possible.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Sharpness Let n = 9k + 6 so that n/3 = 3k + 2 and let |A| = |B| = |C | = n/3.
.. .
.. .
.. .
A
B
C
Figure: Graph with δ = n/3 and n/3 − 2 triangles. Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Question What δ is required for a 2-factor with exactly two cycles?
... K(n−1)/3
...
...
K
(n−4)/3
... K
(n−1)/3
Figure: G with no 2-factor of two cycles, δ =
Ron Gould Emory University
n−1 3 .
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Theorem R.J. Faudree, RG, M.S. Jacobson Let k be a positive integer. If G is a claw-free graph of order n ≥ 2k 4 − 2k 2 + k with δ(G ) ≥ n/k, then G contains a 2-factor with k − 1 components. Further, this value of δ(G ) is best possible.
The case k = 3 say δ ≥ n/3 impies a 2-factor with exactly 2 cycles, which is best possible by the earlier example.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Can we say more about 2-factors? Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of sufficiently large order n with δ(G ) ≥ n/3, then G contains a 2-factor with k disjoint cycles, for 2 ≤ k ≤ ⌊n/3 − 2⌋.
Note: no connectivity condition here! Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Recall a classic result.
Theorem Corradi and Hajnal, 1963 If G is a graph of order n = 3k with δ(G ) ≥ 2n/3, then G contains k disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
The claw-free case, here n = 3k. K 3t+2
... ... ... K 3t+1 Figure: No triangle cover and δ < n/2.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
A claw-free C-H type theorem
Theorem R.J. Faudree, RG, M.S. Jacobson If G is a claw-free graph of sufficiently large order n = 3k with δ(G ) ≥ n/2, then G contains k disjoint triangles.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Sketch of proof
Theorem Li, Rousseau and Zang, 2000 The ramsey number r (Kk , Kn ) ≤ (1 + o(1))
Ron Gould Emory University
nk−1 . (log n)k−2
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Useful Lemma
Lemma In a claw-free graph G with δ(G ) ≥ n/k, and k ≥ 2, then α(G ) ≤ 2k − 1.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
m≈
√
n−ǫ
B2 deg ≥ 17 to C Km
B1 ≥ n/2 − o(n)
remaining vertices i.e.
C = G − (A ∪ B) |C| ≥ n/2 − o(n)
A
B degree at most 16 to A
Ron Gould Emory University
C
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Steps in proof:
1. Find one or two vertices in A ∪ B2 ∪ C with three or more adjacencies into B1 . Since claw-free then we get a triangle using 2 vertices of B1 . 2. We reduce B1 until multiple of 3, then by denseness of it, we apply C-H to get triangle cover of that part.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
Steps in proof:
3. Now as r (K3 , K4 ) = 9, we can place vertices of B2 in triangles until at most 8 remain. Each of these has degree at least 17 into C and can be placed on triangles using C . Now do a similar thing with C (using r = 9) until at most 8 remain. But these 8 have degree at least 17 in A and can be placed on triangles. 4. Finally, what remains of A is a clique and has order a multiple of 3, so we may easily finish.
Ron Gould Emory University
Minimum Degree and Disjoint Cycles in Claw-free Graphs
