Random Partitions and Edge-Disjoint Hamiltonian Cycles

Random Partitions and Edge-Disjoint Hamiltonian Cycles Stephen G. Hartke∗ Department of Mathematics University of Nebraska–Lincoln [email protected]

Tyler Seacrest Department of Mathematics The University of Montana Western t [email protected]

May 3, 2013

Abstract Nash-Williams [21] proved that every graph with n vertices and minimum degree n/2 has at least b5n/224c edge-disjoint Hamiltonian cycles. In [20], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound √ of b(n + 4)/8c. Let α(δ, n) = (δ + 2δn − n2 )/2. Christofides, K¨ uhn, and Osthus [7] proved that for every  > 0, every graph G on a sufficiently large number n of vertices and minimum degree δ ≥ n/2 + n contains α(δ, n)/2 − n/4 edge-disjoint Hamiltonian cycles. Their proof uses Szemer´edi’s Regularity Lemma, and hence the “sufficiently large” requirement on n is a strong condition. In this paper we prove a similar result using methods that do not rely on the Regularity Lemma. In particular, we prove that every graph on n vertices with minip 3/4 mum degree δ ≥ n/2 + 3n ln(n) contains α(δ, n)/2 − 3n7/8 (ln n)1/4 /2 edge-disjoint Hamiltonian cycles. Our proof rests on a structural result that is of independent interest: let G be a graph on n vertices, where n = pq. Then there exists a partition of the p vertices of G into q parts of size p such that every vertex v has at least deg(v)/q − min{deg(v), p} · ln(n) neighbors in each part.

1

Introduction

A classic theorem of Dirac [9] gives a sufficient condition on the minimum degree of a graph for the existence of a Hamiltonian cycle. Nash-Williams [21] strengthened this result, proving Theorem 1 (Nash-Williams [21]). If G is a graph on n vertices and minimum degree at least n/2, then G contains at least b5n/224c edge-disjoint Hamiltonian cycles. ∗

Supported in part by National Science Foundation grant DMS-0914815.

1

Nash-Williams then asked what the best possible result of this kind is. Based on an idea of Babai, Nash-Williams [20] showed an upper bound by constructing graphs with n vertices, minimum degree at least n/2, and at most b(n + 4)/8c edge-disjoint Hamiltonian cycles. Since, the union of k/2 edge-disjoint Hamiltonian cycles is a k-factor, Katerinis [17] studied the size of a largest k-factor, showing that every graph of minimum degree n/2 contains a k-factor for k ≤ (n + 6)/4; Katerinis also gave a construction showing this result is sharp. Egawa [12] improved upon Nash-Williams’ lower bound, achieving n/44 Hamiltonian cycles with the weaker degree condition deg(x) + deg(y) ≥ n for every nonadjacent pair x and y. Frieze and Krivelevich [13] proved that a class of pseudorandom graphs of minimum degree δ = ρn, which includes the Erd˝os–R´enyi random graph G(n, p) with high probability, contain (ρ/2−)n edge-disjoint Hamiltonian cycles for ρ   > 0. Recently, using Szemer´edi’s Regularity Lemma, Christofides, K¨ uhn, and Osthus [7] proved an approximate result for any graph of large minimum degree that is sharp. The following definition will be used in stating their theorem. Definition 2. The function α(δ, n) is given by √ δ + 2δn − n2 α(δ, n) = . 2 Theorem 3 (Christofides, K¨ uhn, and Osthus [7]). For every  > 0, there is a positive integer n0 such that every graph on n ≥ n0 vertices of minimum degree δ ≥ (1/2 + )n contains at least α(δ, n)/2 − n/4 edge-disjoint Hamiltonian cycles. They also showed that this was asymptotically tight. Theorem 4 (Christofides et al. [7]). For all positive integers δ, n with n/2 ≤ δ < n, there is a graph G on n vertices with minimum degree δ such that G contains at most (α(δ, n) + 1)/2 edge-disjoint Hamiltonian cycles. K¨ uhn and Osthus [18] later gave an alternate proof of the theorem as a corollary to another Hamiltonian decomposition result they had proved in a previous paper [6]. This proof too relies on the regularity lemma. A well-known drawback of the Regularity Lemma is that n needs to be extremely large before the lemma applies, at least an exponential tower of 2’s of height proportional to log2 (1/) [15]. We achieve a result similar to that of Christofides et al. using a simpler proof that avoids using the Regularity Lemma. √ Theorem 5. If G is a graph with n vertices and minimum degree δ ≥ n/2+3n3/4 ln n, then G contains α(δ, n)/2 − 3n7/8 (ln n)1/4 /2 edge-disjoint Hamiltonian cycles. Figure 1 shows a graph of α(δ, n)/(2n). Ignoring lower order terms, this graph demonstrates the number of edge-disjoint Hamiltonian cycles achieved by our result as a proportion of the number of vertices of the graph as the minimum degree increases. Our method is similar in spirit to that of Christofides, K¨ uhn, and Osthus, but instead of the partition obtained by the Regularity Lemma we use a partition with different properties given by the following theorem.

2

α(δ, n)/(2n)

1/2

1/8

0.5

0.6

0.7

0.8

0.9

1

δ/n Figure 1: A graph of α(δ, n)/(2n). Theorem 6. Let G be a graph on n vertices, where n = pq. Then there exists a partition of p the vertices of G into q parts of size p such that every vertex v has at least deg(v)/q − min{deg(v), p} · ln(n) neighbors in each part. In addition to proving Theorem 5, Theorem 6 has connections to a conjecture of Bollob´ as and Scott. A bisection is a balanced bipartite spanning subgraph. Bollob´as and Scott conjectured that every graph has a bisection of large degree. Conjecture 7 (Bollob´ as and Scott [2]). Every graph G has a bisection H such that for every vertex v ∈ V (G), bdegG (v)c degH (v) ≥ . 2 Bush [5] proved that every graph on√n vertices contains a bisection where each vertex v has degree at least deg(v)/2 − 4 n ln n, if n is sufficiently large. This proves the leading term in the Bollob´ as–Scott Conjecture is correct. Theorem 6 can be seen as a generalization of Bush’s result, where his statement is recovered and slightly improved upon by setting q = 2. Theorem 6 is also related to discrepancy theory, and in particular multi-colored discrepancy. In multi-colored discrepancy, vertices of a hypergraph H are colored with c colors, so that for every edge e and every color r, a fraction of roughly 1/c of the vertices of e have color r. Theorem 6 can be seen as extending multi-colored discrepancy results to the case where each color class must have equal size. In Section 4, using a multi-colored discrepancy result due to Doerr and Srivastav [10], we improve Theorem 6 in some cases. In particular, in the bisection case of q = 2 we replace the ln(n) factor in the error term by an absolute constant. This work also relates to a conjectured generalization of a result of Kundu [19], who characterized when a graphic sequence π has a realization that contains a k-regular spanning subgraph (such a subgraph is called a k-factor ). Brualdi [3] and Busch, Ferrara, Hartke, Jacobson, Kaul and West [4] independently conjectured that every

3

degree sequence with a realization with a k-factor also has a realization containing k edge-disjoint 1-factors. Partial results by Busch et al. [4] so far have focused on finding as many edge-disjoint 1-factors as possible. By splitting each Hamiltonian cycle into two 1-factors, Theorem 5 gives more edge-disjoint 1-factors than other results in dense graphs. Our paper is organized as follows. In Section 2, we prove Theorem 6 and a useful corollary as well as show the sharpness of this theorem. In light of Theorem 6, it is natural to ask whether the Bollob´as–Scott Conjecture might generalize to multipartite graphs, and in Section 3, we discuss how far such a generalization could be taken. In Section 4, we improve Theorem 6 in some cases using results from discrepancy theory. Before proving Theorem 5 on edge-disjoint Hamiltonian cycles, we first develop several of the key ideas by proving that graphs with high minimum degree have many edge-disjoint 1-factors. Though these results are implied by Theorem 5, the concepts are easier to understand in this simpler setting. In Section 5, we obtain many edgedisjoint 1-factors in graphs on n vertices, where n is a perfect square. In Section 6, we drop the requirement that n be a perfect square. Then in Section 7 we expand these methods to obtain edge-disjoint Hamiltonian cycles, proving Theorem 5. Finally, we conclude with a natural open question.

2

Random Partitions of High Degree

We consider simple graphs, without loops or multiple edges. For any graph G with vertex v, we write N (v) to denote the set of neighbors of v, ∆(G) to denote the maximum degree in G, and δ(G) to denote the minimum degree. The Chernoff–Hoeffding bound is an extremely useful result for showing that a random variable is likely to have values close to its mean. Not only is the bound easy to use, but it gives an exponential drop-off in probability for the random variable to take values further away from the mean. We will use a one-sided version of the Chernoff–Hoeffding bound, which we state below. P Theorem 8 (Chernoff–Hoeffding Bound; see [11]). Let X = ni=1 Xi , where the Xi are independent random variables taking values in the interval [0, 1]. Then Pr[X < E[X] − ] ≤ e

−22 n

.

Using the Chernoff–Hoeffding bound, we show that every graph has a balanced partition of its vertices so that every vertex has many neighbors in each part. This theorem, restated below, will be a key part of the proof of Theorem 5. Theorem 6. Let G be a graph on n vertices, where n = pq for p > 1. Then there exists a partition of p the vertices of G into q parts of size p such that every vertex v has at least deg(v)/q − min{deg(v), p} · ln(n) neighbors in each part. Proof. Let the vertex set be {vij }i≤p,j≤q , where the labeling of the vertices is arbitrary. We think of the vertices as forming the columns and rows of a matrix M . Let R1 , . . . , Rp be the sets of vertices corresponding to the rows M . Let Sq be the symmetric group on q elements. For each row Ri , randomly, independently, and uniformly choose a permutation σi ∈ Sq , and permute the entries of Ri with σi to form a new matrix M 0 . Let C1 , . . . , Cq be the columns of M 0 , each of which has size p.

4

Let Xv,Ci be the random variable indicating the number of v’s neighbors in Ci . We want to calculate the expected value EXv,Ci . We write Xv,Ci as Xv,Ci

=

p X

|Ci ∩ Rj ∩ N (v)|.

j=1

Since each row Ri was permuted independently, Xv,Ci is the sum of independent {0, 1} random variables. Also note we need not include terms of the sum where N (v) ∩ Rj is empty, and hence we can assume Xv,Ci is the sum of at most min{p, deg(v)} such random variables. Let `v = min{p, deg(v)}. We then compute EXv,Ci =

p X

E|Ci ∩ Rj ∩ N (v)| =

j=1

p X |N (v) ∩ Rj |

q

j=1

=

deg(v) . q

√ Let Bv,Ci be the bad event that v has fewer than deg(v)/q − `v ln n neighbors in Ci . Since Xv,Ci is the sum of independent {0, 1} random variables, we can apply the Chernoff–Hoeffding bound, obtaining √

P r[Bv,Ci ] ≤ e−2(

2

`v ln n) /`v

=

1 . n2

Note that there are nq bad events. Applying the union-sum bound, the probability at least one of the bad events occurs is at most nq/n2 , which is less than 1, and hence with nonzero probability C√1 , . . . , Cq forms a partition of the vertices where each vertex v has at least deg(v)/q − `v ln n edges to each part. If n is not exactly pq, we obtain a similar result partitioning G into q parts that are almost equal. Corollary 9. Let G be a graph on n vertices, q a positive integer less than n, p = bn/qc. Then there exists a partition of the vertices ofp G into q parts, each of size p or p+1, such that every vertex v has at least deg(v)/q − min{deg(v), p + 1} · ln(n + q) neighbors in each part. Proof. Form G0 by adding (p + 1)q − n isolated vertices to G. Note that G0 has exactly (p + 1)q vertices. We will now apply Theorem 6 to G0 , but recall that in the proof of that theorem, we arranged the vertices into a matrix arbitrarily. Here we ensure that all the newly added isolated vertices are in the same row. After randomly permuting inside each row, every column will have at most one of these new isolated vertices. By Theorem 6, every vertex v in G0 will have at least p deg(v)/q − min{deg(v), p + 1} · ln((p + 1)q) neighbors in each column of the matrix of vertices. Since each column contains at most one of the isolated vertices, by removing these vertices, we will have a partition p of G into q parts of size p or p + 1, where each vertex has at least deg(v)/q − min{deg(v), p + 1} · ln(n + q) neighbors in each part. The error terms of Theorem 6 and Corollary 9 are not far from being best possible, as the following example shows. The proof below is modified from a result in discrepancy theory; see Spencer [22] and Doerr and Srivastav [10].

5

Theorem 10. For infinitely many n, there exists a graph G on n vertices such that any partitionp of G into q parts contains a part P and vertex v such that v has less than deg(v)/q − 13 n/q 3 neighbors in P . Proof. Let H be a symmetric Hadamard matrix. That is, H is a symmetric matrix with {−1, 1} entries whose columns h1 , . . . , hn are pairwise orthogonal. Symmetric Hadamard matrices are known to exist for all powers of 2 via Sylvester’s construction [16]. By multiplying rows and columns by −1, we can assume that h1 consists entirely of ones. Let A be the corresponding {0, 1} matrix; that is, if J is the matrix with every entry 1, then A = 21 (H + J). Set A = J − A. Let G be the graph whose adjacency matrix is A, with vertices v1 , . . . , vn corresponding to the rows or columns of A. Note that G has some loops, but we will remedy this issue later on. Partition the vertices of G into q parts, and let P be a part not containing v1 with at least b(n − 1)/(q − 1)c vertices. Let χ be a vector of length n with (q − 1)/q in positions corresponding to elements of P , and −1/q everywhere else. Then Aχ is a vector indicating the “discrepancy” of the number of neighbors each vertex has inside of P . That is, the ith component of Aχ is the difference between the number of neighbors of vi inside P and deg(vi )/q. If || · ||∞ is the supremum norm, then ||Aχ||∞ is the maximum of this discrepancy to P over all vertices of G. We will proceed by giving a lower bound for this discrepancy. P First, recall that ||Aχ||∞ ≥ √1n ||Aχ||2 . Also, Aχ = ni=1 12 (χi hi + χi h1 ). If we set P P λ = χ1 + 21 ni=2 χi , then Aχ = λh1 + ni=2 12 χi hi . Thus we can compute ||Aχ||2

v u n n X X u 1 2 1 2 2 2 t χi hi = λ ||h1 ||2 + = λh1 + χi hi 2 2 2 i=2 i=2 2 v v v u n √ uX √ uX n uX 1 2 nu nu (q − 1)2 X 1 2 2 t t t χi hi = χi = + ≥ 2 2 2 q2 q2 i=2

≥ ≥

2

i=2

i∈P

i∈P /

  √ s n (n − 1)(q − 1)2 n−1 1 + n−1− 2 q 2 (q − 1) q − 1 q2 s √ n (n − 1)(q − 1) . 2 q2

q Therefore, there exists a vertex v with fewer than deg(v)/q − 21 (n−1)(q−1) neighbors q2 q in a part P or more than deg(v)/q + 21 (n−1)(q−1) neighbors in some part P . If the q2 former case holds, then the result follows, so assume some vertex v has more than q deg(v)/q +

1 2

(n−1)(q−1) q2

neighbors in P . q Since v has at least deg(v)/q + 21 (n−1)(q−1) neighbors in P , then v has at most q2 q (n−1)(q−1) q−1 1 neighbors in the other q − 1 parts. By the Pigeonhole q deg(v) − 2 q2 q Principle, there is some part with at most 1q deg(v) − 12 qn3 neighbors of v.

If we wish G to be a simple graph, we canqremove loops, and we still have that every vertex v has at most (deg(v) + 1)/q − 12 qn3 neighbors in some part, which is

6

bounded by deg(v)/q −

3

1 3

q

n q3

for large enough n.

A Generalized Bollob´ as–Scott Conjecture

Setting q = 2 in Theorem 6 yields an approximate version of the Bollob´as–Scott Conp jecture: every graph G has a bisection H where degH (v) ≥ degG (v)/2 − deg(v) ln n for every vertex v. Thus, a natural question is whether the Bollob´as-Scott Conjecture is true in more generality. Question 11. Given a positive integer n, for which values q is it true that for every graph G on n vertices there exists a balanced q-partite spanning subgraph H such that every vertex v has bdegG (v)/qc neighbors in each part? The value q = 2 corresponds to the Bollob´as–Scott Conjecture. Values on the order √ of n would be useful for the applications in Sections 5, 6, and 7. We observe that the desired subgraph does not always exist when q = cn for c ∈ (0, 1). Proposition 12. Fix c ∈ (0, 1) such that 1/c is an integer. Let n be a sufficiently large positive integer such that q = cn is an even integer. Then there exists a graph G such that every balanced q-partite spanning subgraph has a vertex v with fewer than bdegG (v)/qc neighbors in some part. Proof. Let G be a random graph where each possible edge appears independently with p probability σ = c + ln n/n. We consider partitions P = {P1 , . . . , Pq }, where each part has size 1/c. The Chernoff–Hoeffding bound shows that the probability a vertex has fewer than √ 2 −2 n ln n /(n−1) , which is at most 1/n2 . Applying the union sum q neighbors is at most e bound, the probability any vertex has fewer than q neighbors is less than 1/n. Thus every vertex has at least q neighbors with probability 1 − 1/n. We next show that there is some vertex v and some part Pi such that v has no neighbors in Pi . Given a partition P, let XP,Pi ,v be the event that v has at least 1 neighbor in Pi . The probability XP,Pi ,v occurs is τ = 1 − (1 − σ)1/c . Note that for a fixed P, the events q/2

AP = {XP,Pj ,v }

for v ∈

[

Pi and j = q/2 + 1, . . . , q,

i=1

are independent, since they rely on disjoint sets of edges. Hence, the probability all 2 cn2 /4 events in AP occur is τ cn /4 . The number of such partitions P is n! ≤ nn = (eln n )n = en ln n . [(1/c)!]cn By the union sum bound, the probability there is a partition that satisfies all the events in AP is at most 2 2 2 en ln n τ cn /4 = en ln n+ln(τ )cn /4 = en ln n+ξn , where ξ = ln(τ )c/4. Thus, the probability that G is such that every vertex has degree at least q and every partition has a part Pi and a vertex v with no neighbor in Pi 2 is at least 1 − 1/n − en ln n+ξn . Since τ < 1, ξ is a negative constant, and hence for sufficiently large n, the probability is positive. Therefore there exists such a graph G.

7

4

Using Results from Discrepancy Theory

The Bollob´ as–Scott Conjecture is related to the well-studied question of discrepancy. Given a collection of subsets of {1, . . . , n}, the goal is to color the numbers in {1, . . . , n} red and blue so that each subset has roughly the same number of blue elements as red elements. To avoid confusion over the meaning of n later on, we will change our ground set from the usual {1, . . . , n} to {1, . . . , t}. Thus, more precisely, given a set A ⊆ {1, . . . , t} and a function σ : {1, . . . , t} → {−1, 1}, the discrepancy of A is X disc(A) = σ(a) . a∈A

Given a collection of t sets S = {A1 , . . . , At }, where each Ai ⊆ {1, . . . , t}, we wish to minimize the maximum discrepancy of√any set in S. A simple probabilistic argument yields that a discrepancy of at most t ln t is achievable. Spencer [22] was able to remove the ln t factor and replace it with an absolute constant, and furthermore showed this constant could be as low as 6. When the number of parts q in Theorem 6 is 2, then it can be seen as extending the discrepancy result where the color classes on {1, . . . , t} must have equal size. Doerr and Srivastav [10] extended many discrepancy results to multi-colored discrepancy. Given a collection of subsets of {1, . . . , t}, the goal is to color the numbers in {1, . . . , t} with many colors so that no subset has too many or too few elements that are a given color. More precisely, in c-colored discrepancy, we start with a coloring function σ : {1, . . . , t} → {1, . . . , c}. For every color i ∈ {1, . . . , c} we have a function σi given by  c−1 if σ(a) = i, c σi (a) = − 1c if σ(a) 6= i. Then the discrepancy of a set A ⊆ {1, . . . , t} relative to color i is given by X disc(A, i) = σi (a) . a∈A

Note that this definition is natural in that a set A with |A|/c elements colored red has a red-discrepancy of zero, and if we increase or decrease the number of red elements by a fixed amount `, then the red-discrepancy increases by `. Again, given a collection of m sets H = {A1 , . . . , Am }, the goal is to minimize the maximum discrepancy over all sets in H and all colors, yielding a quantity disc(H, c): disc(H, c) = min σ

max

A∈H,1≤i≤c

disc(A, i)

Extending the simple probabilistic qargument for the two color case, Doerr and Srivastav [10] showed that disc(H, c) ≤ 12 t ln(mc). Going farther, they proved the following bound that was analogous to the result by Spencer. Theorem 13 (Doerr and Srivastav [10]). r disc(H, c) ≤ O

8

! t  mc  ln . c t

We can use this to obtain a result about balanced partitions of high degree. Theorem 14. There exists an absolute constant K with the following property: Let G be a graph on n vertices, where n = pq for p > 1. Then there exists a partition of thepvertices of G into q parts of size p such that every vertex v has at least deg(v)/q − K n ln(q) neighbors in each part. Proof. Form the matrix M from the proof of Theorem 6. Recall this matrix has p rows and q columns. Let Av,i be the rows containing vertices from N (v) ∩ Ci . Then there are qn sets of rows Av,i . Setting H = {Av,i }, t = p, c = q, and m = qn, we apply Theorem 13, and obtain σq of the rows q a coloring   of M with q colors where the p p 3 discrepancy is at most O =O q ln q q ln q . If Ri receives color j under σ, perform a cyclic shift of the row Ri , where the shift is j units to the right. This forms a new matrix M 0 . Let C10 , . . . , Cq0 be the columns of M 0 . Now consider a vertex v and column Cj0 . How many neighbors does v have in Cj0 ? To find out, we will first consider the question of how many neighbors in Cj0 were originally in Ci ∩ N (v). A vertex u in Ci ∩ N (v) moved to Cj0 if u’s row was colored so that it moved from Ci to Cj0 . Suppose color ` is required to make this change. Thus, we need to find out how many vertices in Ci ∩ N (v) had a row colored `. Notice that Av,i are exactly the rows corresponding in Ci ∩ N (v). We  q to vertices p q

ln q . Therefore, the number  q p ln q . In neighbors of v that move from Ci to Cj0 has a discrepancy of at most O q q  p other words, we have at least |Av,i |/q − O q ln q neighbors of v that started in Ci know Av,i with color ` has discrepancy at most O

end up in Cj0 . Summing over all i, the total number of neighbors of v that end up in   q Cj0 is at least deg(v)/q − O q pq ln q , which is deg(v)/q − O(n ln q), as desired. Note that this theorem is an improvement over Theorem 6 when q is a constant relative to n.

5

Edge-Disjoint 1-Factors

In this section, we develop a general schema for using Theorem 6 to find edge-disjoint 1-factors and Hamiltonian cycles in graphs of high minimum degree. We also apply it in the simplest setting: finding many edge-disjoint 1-factors in a graph on n vertices where n is a perfect square. We first present these key ideas in the simpler setting of 1-factors before using them to obtain Hamiltonian cycles so that technical details do not obscure the big picture. We use the following result of Csaba. Theorem 15 (Csaba [8]). Let G be a simple balanced bipartite graph on 2p vertices with minimum degree δ at least p/2. Then G has a bα(δ, p)c-regular spanning subgraph. Using the following classical theorem, we can split the regular bipartite graph from Csaba’s Theorem into 1-factors. Theorem 16 (Marriage Theorem, see [23]). Every r-regular bipartite graph decomposes into r edge-disjoint 1-factors.

9

We also use another classical theorem. Theorem 17 (See [23]). For q even, the complete graph on q vertices decomposes into q − 1 edge-disjoint 1-factors. We now prove a result showing that a graph of high minimum degree contains many edge-disjoint 1-factors. We will outline the general schema for proving this result, which will be followed again later as we improve upon this result in Sections 6 and 7. Schema 18 (Finding Edge-Disjoint 1-Factors). Let G be a graph of high minimum degree. 1. Using our partition theorem, we form a partition P1 , . . . , Pq such that every vertex has many neighbors in each part. We think of P1 , . . . , Pq as forming the vertices of a complete host graph H. Note that H is isomorphic to Kq . 2. We form a set of edge-disjoint 1-factors in H. We then “multiply” each of the 1-factors to create a multiset F of 1-factors of H, where no edge of H appears in too many 1-factors. 3. Given a pair of parts Pi and Pj , we apply Csaba’s Theorem (or extensions) and the Marriage Theorem to the induced bipartite subgraph between Pi and Pj to obtain edge-disjoint perfect matchings between Pi and Pj . 4. Using the perfect matchings between each pair of parts, each 1-factor of F is “blown up” into a 1-factor of G. The simplest use of this schema yields the following theorem. Theorem 19. Let G be a graph √ on n vertices, where n =pq and q √ is even. If G has minimum degree δ at least n/2 + q p ln n, then G contains α(δ/q − p ln n, p) (q − 1) edge-disjoint 1-factors. Proof. We follow Schema 18.

Step 1: Using Theorem 6, we partition the vertices of G into√q equal parts P1 , . . . , Pq ,

each of size p, so that every vertex has degree at least δ/q − p ln n to each part. We think of P1 , . . . , Pq as being vertices in a host graph H isomorphic to Kq . 

√



Step 2: Set k = α(δ/q − p ln n, p) . Since q is even, by Theorem 17 H decomposes into (q − 1) 1-factors. Duplicating each of these 1-factors k times yields a multiset F of k(q − 1) 1-factors in H, where each edge is used k times. √

Step 3: Since δ ≥ n/2 + q p ln n, the induced bipartite subgraph between each pair of parts has minimum degree at least √ n/2 + q p ln n p − p ln n = p/2. q Hence the bipartite subgraph between each pair of parts is dense enough to apply Csaba’s Theorem. Applying Csaba’s Theorem, we obtain an k-regular spanning subgraph between each pair of parts. By the Marriage Theorem, each regular subgraph between a pair of parts decomposes into k edge-disjoint perfect matchings.

10

Figure 2: The union of matchings between parts is a 1-factor of G. Here we have q = 6 and p = 3. Step 4: Let e be an edge in H with endpoints Pi and Pj . From Step 3, there are k edge-disjoint perfect matchings in G between Pi and Pj , and from Step 2, there are k 1-factors in F that contain e. Arbitrarily create a bijective assignment from the matchings in G between Pi and Pj to the 1-factors in F containing e. Each F ∈ F corresponds to a 1-factor of G obtained by taking the union of the matchings assigned to it. See Figure 2. Thus, we have k(q − 1) edge-disjoint 1-factors in G. Setting q = 2 and p = n/2, we see that √ every graph on an even number n of vertices of minimum degree δ at least n/2 + 2n ln n has at least α(δ, n)/2 −√n3/4 ln n edge-disjoint 1-factors. (To simplify this lower bound, we use the inequality a − b ≥ √ √ a − b when a ≥ b ≥ 0.) Theorem 19 is more powerful if p and q are both large. For √ example, if n is an even perfect square and we set p = q = n, then there exists at least α(δ, n) − 4n3/4 ln n edge-disjoint 1-factors in a graph on n vertices of minimum degree δ ≥ n/2 + n3/4 ln n, an asymptotic improvement by a factor of 2. When n is a perfect square, this bound is asymptotically tight due to Theorem 4. In the next section, we obtain asymptotically tight results for when n is not a perfect square.

6

Asymptotically Sharp Number of 1-Factors

As a consequence of Theorem 19, we obtain nearly α(δ, n) many edge-disjoint 1-factors when n is a perfect square and δ is large. In this section, we remove the requirement that n is a perfect square. The machinery we develop will also be used later to obtain edgedisjoint Hamiltonian cycles. We follow Schema 18, but instead of applying Theorem 6, we use Corollary 9 to partition the graph. We also need several modifications to account for the fact that not all of the parts are the same size. √ Despite n not being a perfect square, we still need to choose p and q close to n before applying Step 1 of Schema 18. Furthermore, we need that in our partition of the vertices into q parts there are relatively few parts of size p + 1. √ Lemma 20. Given any positive integer n, there exists an integer p where p ≥ n such that if q = bn/pc, then • q is even,

11

√ √ • p − n < 2n1/4 + 4, √ √ • n − q < 2n1/4 + 4, and √ • n − pq < 4 2n1/4 + 8. In particular, a set √ of size n can be partitioned into parts of sizes p and p + 1 such that there are at most 4 2n1/4 + 8 parts of size p + 1. √ Proof. Let t = d ne. Let s2 be the smallest perfect square such that t2 −s2 ≤ n and t−s √ is even. Set p = t+s and q = t−s. We know that t < n+1. We also have (t−1)2 ≤ n, which implies t2 − n ≤ 2t − 1. We also know (s − 2)2 is too small for t2 − − 2)2 to be p(s√ √ less than n, hence (s − 2)2 ≤ t2 − n ≤ 2t − 1. Hence p s√≤ 2t − 1 + 2√< 2 n + 1 + 2. √ √ Both p − n and n − q are at most s + 1 < 2 n + 1 + 3 ≤ 2n1/4 + 4. Since pq = t2 − s2 is less than n, but t2 − (s√ − 2)2 is greater than n, we see n − pq is at most 4s − 4, and hence n − pq is at most 4 2n1/4 + 8. In Step 3 of Schema 18 we need to find many edge-disjoint matchings between parts of size p and p + 1 that saturate the part of size p. To do so, we apply the following extension of Csaba’s Theorem. To state the result, we will use the following terminology. Given a function f : V (G) → N, an f -factor of G is a spanning subgraph H such that degH (v) = f (v) for all vertices v. Given a matching M of G and a subset Y of vertices, we say M saturates Y if every vertex of Y is incident to an edge of M . Lemma 21. Let G be a bipartite graph on X ∪ Y with |X| = p + 1, |Y | = p, such that vertices in X have minimum degree δ, and vertices in Y have minimum degree δ + 1 for δ ≥ p/2. Let k ≤ α(δ, p) and let f : V (G) → N satisfy f (v) ≤ k, if v ∈ X, f (v) = k, if v ∈ Y , P P and v∈X f (v) = v∈Y f (v). Then G has an f -factor H. Furthermore, H decomposes into matchings that saturate Y , where, for each v ∈ X, k − f (v) of the matchings avoid v. Proof. Suppose the lemma is not true. Then there are some functions f satisfying the requirements of the lemma such that G has no f -factor. Among these, choose the f to minimize the number of vertices x ∈ X such that f (x) < k. Order the vertices of X = {x0 , . . . , xp } such that f (x0 ) ≥ f (x1 ) ≥ · · · ≥ f (xp ). Choose a b-factor B of G and an integer t such that • b(y) = k for y ∈ Y , • b(xi ) = f (xi ) for i < t, • b(xt ) > f (xt ), • t is maximized, and • (after maximizing t) the quantity b(xt ) − f (xt ) is minimized for the given t. Note that by Csaba’s Theorem, we can find such a b-factor for b(xi ) = k for i < p and b(xp ) = 0, and hence b is well-defined.

12

Figure 3: This is a visualization of the three sequences f , b, and h. For i = 0, 1, 2, . . . , p, the height of the three lines represents f (xi ), b(xi ), and h(xi ) respectively. Note that f (xt ) is between the two values b(xt ) and h(xt ). This allows us to modify b to form b0 so that b0 (xt ) is closer to f than b is, contradicting the choice of b. Define the function h on the vertices of G by   if i < t f (xi ) h(xi ) = k if i > t  Pp  f (xt ) − i=t+1 (k − f (xi )) if i = t Pp and P h(y) = k for y ∈ Y . Since i=0 f (x Pip) = kp and f (xi ) ≤ k for all xi , we have f (x ) ≥ k(p − t). Thus f (x ) ≥ i t i≥t i=t+1 (k − f (xi )), and hence h(xt ) is nonnegative. Let H be a h-factor of G. Such an h-factor exists, since it has fewer non-k entries than f . See Figure 3. Let D(B) be the collection of edges in B but not in H, and let D(H) be the collection of edges in H but not in B. Let P be a maximal alternating trail starting on an edge in D(B) incident to xt , and then alternating between edges in D(B) and edges in D(H). Note that such an edge exists since b(xt ) > h(xt ). This is an alternating trail in the graph D(B) ∪ D(H). In this graph, vertices in Y have equal b-degree and h-degree, so P does not end in Y . Nor does P end on x0 , . . . , xt−1 , since each of these also have equal number of incident B-edges and H-edges. Hence P ends on xi for i > t. Switching along this alternating path modifies B into a new graph B 0 , which is a b0 -factor such that • b0 (y) = k for y ∈ Y , • b0 (xi ) = f (xi ) for i < t, • b0 (xt ) ≥ f (xt ), and • the quantity b0 (xt ) − f (xt ) is smaller than b(xt ) − f (xt ), which is a contradiction of the choice of b. Now we know there exists such an f -factor F . We now wish to partition F into matchings that saturate Y . For any vertex x ∈ X such that f (x) < k, consider the

13

graph F 0 consisting of H − x. Let Y 0 ⊆ Y , and consider N = NH 0 (Y 0 ). In H 0 , there are at least |Y 0 |k − f (x) edges leaving Y 0 . The set N has at most k|N | edges entering it. If |N | < |Y 0 |, then Y 0 has at least k − f (x) more edges leaving than N has entering, a contradiction. Hence |N | ≥ |Y 0 |, and by Hall’s Matching Theorem, there exists a matching saturating Y that avoids x. We can keep removing matchings in this way until H is totally partitioned. Applying Lemma 21 in Step 3 leaves one vertex unmatched in the part of size p + 1 in each matching. To match these extra vertices in a 1-factor of G, we need extra edges between the parts of size p + 1. As in Theorem 19, we “blow-up” a 1-factorization of H into many edge-disjoint 1-factors in G. However, in this case we need to throw out a small number of 1-factors of H that include edges between parts of size p + 1. The following lemma tells us that there are enough remaining edge-disjoint 1-factors in H. Lemma 22. Let q be even, and let A be a set of vertices in Kq with |A| = a ≥ 2. Then there exist at least q − 2a + 2 edge-disjoint 1-factors in Kq with no edges completely contained inside of A. Proof. A standard 1-factorization of Kq proving Theorem 17 is constructed by drawing Kq in the plane by placing q − 1 of the vertices as the vertices of a regular (q − 1)polygon, and placing the last vertex at the center. Edges thus become line segments, each of which has a slope. The set of all edges with a fixed slope s, plus the one edge from the center vertex with perpendicular slope, form a 1-factor. 1-factors of this type partition the edges, so this forms a 1-factorization. Let F be this set of (q −1) 1-factors. Without loss of generality, assume the vertices of A are consecutive vertices along the polygon, v1 , . . . , va . We form Fˆ by removing from F any 1-factor with an edge inside of A. An example of the 1-factors that must be removed is given in Figure 4. How many 1-factors have we removed? Suppose F is a 1-factor that was removed. Let vi vj be an edge of F that is inside of A. Note that if |i − j| > 2, then we can find another edge vi+1 vj−1 of F that is also inside A, since the edges vi vj and vi+1 vj−1 have the same slope. Thus, we can assume that |i − j| = 1 or |i − j| = 2. There are only a − 1 such edges if |i − j| = 1, and a − 2 such edges if |i − j| = 2. Therefore, there are at most (2a − 3) 1-factors removed since every removed 1-factor has at least 1 such edge vi vj . Note that if a ≥ q/2 + 2, then there may be two such edges vi vj in a single 1-factor, and hence we only obtain a bound on the number of removed 1-factors. Thus Fˆ contains at least (q − 1) − (2a − 3) = q − 2a + 2 1-factors. We now put these ingredients together. Theorem 23. Let G be a graph on n vertices, where n is even. If G has minimum √ 3/4 degree δ at least n/2 + 3n ln n, then G contains at least α(δ, n) − 3n7/8 ln(n) edgedisjoint 1-factors. Proof. We again follow Schema 18.

Step 1: Choose p and q as given by Lemma √ 20. Hence, p and q are both within √ √

2n1/4 + 4 of n, q is even, and pq is within 4 2n1/4 + 8 of n. Note that the theorem is vacuously true for n ≤ 160, √ 000 since for smaller values of n, the minimum degree requirement of n/2 + 3n3/4 ln n cannot be achieved by a simple graph. Hence we assume n ≥ 160, 000, and thus q is positive. Let P1 , . . . , Pq be a partition of the

14

Figure 4: A standard 1-factorization of Kq consists of matchings like the solid one shown, rotated around the figure, such as the lighter, dashed matchings. The three matchings shown are the ones that are removed in Lemma 22 since they each have an edge contained in set A.

Pj

Pi A

B P i0

Figure 5: Matching from parts in A to parts in B. vertices of G given by Corollary 9. Let A be √ the set of parts of size p + 1, and B be the parts of size p. By Lemma 20, |A| ≤ 4 2n1/4 + 8. Let H be the complete graph on vertex set A ∪ B.

Step 2: If A is empty, then by Theorem 17, then H decomposes into (q−1) 1-factors. If A is non-empty, by Lemma 22, there exists a set of 1-factors in H with q − 2|A| + 2 edge-disjoint 1-factors with no edges in A. In either case, we have a set of at least √ q − 8 2n1/4 − 14 edge-disjoint 1-factors with no edges in A. Set j k p k = α(δ/q − (p + 1) ln(n + q), p) . By duplicating each 1-factor k times, we have a multiset F of 1-factors in H, where each edge √ in H is used at most k times, and no edge within A is used. Hence |F| ≥ (q − 8 2n1/4 − 14)k.

15

Step 3: By Corollary 9, the minimum degree of the induced bipartite subgraph p between every pair of parts is at least δ/q − (p + 1) ln(n + q). Note that for n ≥ 3, p p q (p + 1) ln(n + q) + q = (pq · q + q 2 ) ln(n + q) + q q √ √ (n n + n) ln(n2 ) + n ≤ p ≤ 3n3/4 ln(n),

where the second line uses the bounds from Lemma 20. Hence p √ δ ≥ n/2 + 3n3/4 ln n ≥ n/2 + q (p + 1) ln(n + q) + q, and the minimum degree of the induced bipartite subgraphs is at least p n/2 + q (p + 1) ln(n + q) + q p − (p + 1) ln(n + q) ≥ p/2 + 1. q Thus we can apply Csaba’s Theorem and Lemma 21. In each induced bipartite subgraph between a pair of parts in B, by Csaba’s Theorem there is a regular spanning subgraph of degree k (as defined in Step 2), which decomposes into k perfect matchings. We next handle the parts in A, with our aim to create matchings like those shown in Figure 5. Note that |A| is even, since n and q are even and |A| = n − pq. Thus we can form an arbitrary matching M of the parts in A. Fix twop parts Pi and Pi0 matched in M . By counting degrees, there are at least (p + 1)(δ/q − (p + 1) ln(n + q)) edges between Pi and Pi0 . Using bounds on p and q from Lemma p 20 and n ≥ 160, 000, the number of edges between Pi and Pi0 is at least δ − 2n3/4 ln(n) since   p p (p + 1) δ/q − (p + 1) ln(n + q) ≥ δ − 2n3/4 ln(n) (1) p (see Section 9.1 for a detailed calculation). We may assume that |F| ≤ δ −2n3/4 ln(n) since otherwise we could reduce the size of F by lowering k, as long as |F| ≥ α(δ, n) − √ 3n7/8 (ln n)1/4 (note that reducing k by 1 reduces the size of F by at most q − 1 ≤ n). Thus there are at least as many edges between Pi and Pi0 as there are 1-factors in F. For each Pi Pi0 ∈ M , we create an injective map φPi ,Pi0 from F to the edges between Pi and Pi0 . Let e be an edge of H between Pi ∈ A and Pj ∈ B. We need matchings between Pi and Pj that saturate Pj . For v ∈ Pi , let Φe,v Pi ,P 0 = {F ∈ F : e ∈ F and v is an endpoint of φPi ,Pi0 (F )}, i

S where Pi and Pi0 are matched in M . Note that v∈Pi Φe,v Pi ,Pi0 has size k since there are k 1-factors in F that contain e. The target f -factor for Lemma 21 applied to Pi and Pj is constant k on Pj and for v ∈ Pi , we set f (v) = k − Φe,v Pi ,P` . S Since | v∈Pi Φe,v Pi ,P` | = k, the sums of f (v) over Pi and over Pj are equal. Hence we can apply Lemma 21 to obtain matchings that saturate Pj , where |Φe,v Pi ,Pi0 | of the matchings miss v ∈ Pi .

16

Step 4: Let e be an edge in H with endpoints Pi and Pj . Step 4.I: e is contained within B. From Step 3, there are k edge-disjoint matchings between Pi and Pj , and from Step 2, there are k 1-factors in F that contain e. Arbitrarily create a bijective assignment from the matchings in G between Pi and Pj to the 1-factors in F containing e.

Step 4.II: e is between A and B. We assume endpoint Pi is in A and Pj is in B.

Fix v ∈ Pi . From Step 3, there are |Φe,v Pi ,Pi0 | matchings between Pi and Pj that avoid v, e,v and the set ΦPi ,P 0 contains the 1-factors F in F that contain e and where φPi ,Pi0 (F ) i has v as an endpoint. Arbitrarily create a bijective assignment from the matchings between Pi and Pj that avoid v to the 1-factors of Φe,v Pi ,P 0 . i

Each F ∈ F corresponds to a 1-factor of G obtained by taking the union of the matchings assigned to it, along with {φPi ,Pi0 (F ) : Pi and Pi0 are matched in M }. We √ √ thus have |F| 1-factors G, where |F| ≥ (q − 8 2n1/4 − 14)k = qk − (8 2n1/4 + 14)k. √ of1/4 We bound qk and (8 2n + 14)k separately. To do so, we will make frequent use of √ √ √ the inequality x − y ≥ x − y for x ≥jy ≥ 0. k p First we bound qk. Recall that k = α(δ/q − (p + 1) ln(n + q), p) . Using the p bound (p + 1) ln(n + q) ≤ 4n1/4 ln n and the bounds on p, q, and n − pq from Lemma 20, we have q √ qk ≥ α(δ, n) − 2δ( 2n1/4 + 2) + 2n7/4 ln(n) − 2n3/4 ln(n). We can then use δ ≤ n and n ≥ 160, 000 to show qk ≥ α(δ, n) − 1.42n7/8 (ln n)1/4 ,

(2)

described in detail in Section √ 9.2. √ We also need to bound (8 2n1/4 + 14)k. Here, we use the bound k ≤ p ≤ n + √ 1/4 2n + 8 and n ≥ 160, 000, and we see √ (8 2n1/4 + 14)k ≤ 1.57n7/8 (ln n)1/4 (3) (see Section 9.3 for details). Combining Inequalities 2 and 3, we achieve α(δ, n) − 3n7/8 (ln n)1/4 edge-disjoint 1-factors.

7

Edge-Disjoint Hamiltonian Cycles

We now find many edge-disjoint Hamiltonian cycles in a graph of high minimum degree. We again follow Schema 18, but instead of finding 1-factors of the host graph H, we use Hamiltonian cycles of the host graph H, which we translate to Hamiltonian cycles of the larger graph G. We need to find Hamiltonian cycles in the host graph H in 1-to-1 correspondence with the Hamiltonian cycles we will obtain in G. However, simply multiplying a set of edge-disjoint Hamiltonian cycles in H will not work. As in Section 6, we will need

17

the Hamiltonian cycles to not have edges completely contained in part A in order to match up leftover vertices in parts of size p + 1. Also, we will need each Hamiltonian cycle to go through a new collection of vertices C, but with each edge used a very few number of times. Part C will be used to modify the collection of two-cycles we obtain to form Hamiltonian cycles. Finally, we also need no edge of H be used too many times. Lemma 27 produces a set of Hamiltonian cycles of H that satisfy all these requirements. Theorem 24 (See [1]). The complete graph on an odd number b ≥ 3 of vertices decomposes into (b − 1)/2 edge-disjoint Hamiltonian cycles. In the proof of Lemma 27, we use a multiplicative form of the Chernoff-Hoeffding bound. P Theorem 25 (Chernoff–Hoeffding Bound; see [11]). Let X = i∈[n] Xi , where the Xi are independent random variables taking values in the interval [0, 1]. Then for 0 < δ < 1, 2 Pr(X > (1 + δ)µ) < e−µδ /3 . Definition 26. Let H be a graph and F a multiset of Hamiltonian cycles of H. The multiset F need not contain edge-disjoint cycles, and in fact may contain multiple copies of identical Hamiltonian cycles. Given an edge e in H, we define NH (F, e) to be the number of Hamiltonian cycles in F containing e. We write N (F, e) when the graph H is clear from context. For a graph H and a subset X of vertices, recall that we say an edge e is within X if both endpoints of e are in X. √ Lemma 27. Let q, a, b, c be nonnegative integers with q ≥ 367, q = a+b+c, a ≤ 10 q, b odd, and q 3/4 ≤ c ≤ q 3/4 + 2. Let A, B, C be a partition of V (Kq ) such that |A| = a, |B| = b, and |C| = c. Then for any 0 ≤ τ ≤ q, there exists a multiset F of at least τ q/2 − 7q 7/4 Hamiltonian cycles of Kq such that √ 1. for every edge e in Kq , N (F, e) ≤ τ + 9q ln q, 2. for e within A, N (F, e) = 0, √ 3. for e within C, N (F, e) ≤ q, 4. and every F ∈ F has exactly one edge within C. Proof. By Theorem 24, there is a set B of (b − 1)/2 edge-disjoint Hamiltonian cycles that span B. We form the multiset D by duplicating each cycle of B exactly τ times. √ Since b = q − a − c ≥ q − 10 q − q 3/4 − 2 and q ≥ 367, then D is a set of at least τ (b − 1)/2 ≥ τ q/2 − 0.53q 7/4

(4)

cycles that span B (see Section 9.4 for details). We just need to extend these cycles to cover the entire graph Kq . We assign to each cycle F ∈ D an edge eF within C so that no edge within C is used √ more than q times. We have enough edges within C to make this assignment, since √ there are at least (q 3/2 − q 3/4 )/2 edges in C. We may assume |D| ≤ [(q 3/2 − q 3/4 )/2] q, since otherwise we could remove cycles from D as long as |D| ≥ τ q/2 − 7q 7/4 .

18

Figure 6: A Hamiltonian cycle in Kq formed by extending a cycle F spanning B. Fix a cycle F ∈ D, and let u and v be the endpoints of eF . We will extend F to cover the vertices in A ∪ C using labels on the edges of F to guide the extension. Let LF = A ∪ (C − {u, v}) ∪ {eF } be the label set of F . Let φF be an arbitrary injection from LF to {0, 1, · · · , b − 1}. Choose a random variable iF that takes values uniformly from {0, 1, . . . , b − 1} and independently for each F . Finally, define φ0F such that for a ∈ LF , φ0F (a) = φF (a) + iF (mod b). Let {e0 , . . . , eb−1 } be the edges of F . Then we extend F by using edge eφ0 (a) to take a detour through a ∈ LF as in Figure 6. In particular, for a vertex a ∈ LF , we remove edge eφ0 (a) = xy and replace it with the edges xa and ay. For the edge eF ∈ LF , we remove eφ0 (eF ) = xy and replace it with the edges xu, uv, and vy. Extend each cycle F ∈ D independently in this way to create a new multiset F of Hamiltonian cycles of Kq . The Hamiltonian cycles in F satisfy conditions 2, 3, and 4 of the theorem statement by construction, and satisfy condition 1 for all edges except perhaps the edges between A ∪ C and B. We show that there is an assignment of the random variables iF so that F satisfies condition 1 for these edges as well. Let F be a fixed cycle in D and let F 0 be its extension in F. For an edge e = xy with x ∈ A ∪ C and y ∈ B, the probability e is in F 0 is 2/b if x 6= u, v and is 1/b if x = u or x = v, where eF = uv. Note that N (F, e) is a random variable counting the number of Hamiltonian cycles in F that contain e. If we set µ = EN (F, e), we see µ ≤ 2b |F| = b−1 b τ < τ . For a given 0 edge e, the event that F ∈ F contains e is independent of the event that another cycle in F contains e.√For every edge e between A ∪ C and B, define a bad event to be when N (F, e) ≥ τ + 6q ln q. We want to show that the probability of these bad events is

19

low. Applying Theorem 25 and since µ ≤ τ ≤ q, we have     p p Pr N (F, e) ≥ τ + 9q ln q ≤ Pr N (F, e) ≥ µ + 9q ln q   p ≤ Pr N (F, e) ≥ µ + 9µ ln q     p = Pr N (F, e) ≥ 1 + 9 ln q/µ µ √ 2 ≤ e−µ·( 9 ln q/µ) /3 1 ≤ . q3 √ Since 0 ≤ a ≤ 10 q, q 3/4 ≤ c ≤ q 3/4 + 2, we have at most √ b(a + c) ≤ (q − q 3/4 )(10 q + q 3/4 + 2) edges between A ∪ C and B. By the union-sum bound, the probability any bad event √ occurs is at most 1/q 3 times (q − q 3/4 )(10 q + q 3/4 + 2). This value satisfies 1 √ (q − q 3/4 )(10 q + q 3/4 + 2) < 1. 3 q

(5)

This is strictly less than 1 for values of q ≥ 6 (see Section 9.5 for details). Consider a Hamiltonian cycle F on H minus an edge e between parts Pi and Pj in C from our random partition. We will expand F to a collection of paths that pass through all the vertices, matching the vertices of Pi to Pj . At that point, we will need to find a perfect matching between Pi and Pj that completes F into a Hamiltonian cycle. Lemma 29 allows us to do this. To prove it, we use the following generalization of Dirac’s Theorem due to Ghouila-Houri [14]. Theorem 28 (Ghouila-Houri [14]). Let D be a digraph on n vertices such that the minimum in-degree plus the minimum out-degree is at least n. Then D contains a directed Hamiltonian cycle. By looking at the split of the digraph, it is equivalent to the following result. Lemma 29. Let G be a bipartite graph with each part of size p and minimum degree p/2 + 1, and let M be a perfect matching of G. Then M can be extended to a Hamiltonian cycle. Proof. Suppose G has bipartition L ∪ R. For each vertex x ∈ L, let x0 be the vertex in R that is matched with x in M . Let each xx0 ∈ M form the vertices of a digraph D. For each edge xy 0 ∈ G for y 0 6= x0 , place an edge from xx0 to yy 0 in D. We have chosen D so that G is the split of the digraph D. Note that D is a digraph on p vertices and each vertex has in-degree at least p/2 and out-degree at least p/2. Applying the Ghouila-Houri Theorem to D, we obtain a directed Hamiltonian cycle, where the vertices of this Hamiltonian cycle, in order of the cycle, have the form x1 x01 , x2 x02 , . . ., xn x0n . Since there is a directed edge from xi x0i to xi+1 x0i+1 in D, we have an edge between xi and x0i+1 in G. Hence the edges x1 x01 , x1 x02 , . . ., xn x0n , xn x01 form a Hamiltonian cycle in G.

20

Figure 7: G is partitioned into sets of size p or p + 1, and these parts make up three sets A, B, and C as shown. Now we are ready to prove the main result. Theorem 5. Let G be a graph on n vertices. If G has minimum degree δ at least n/2+ p 7/8 1/4 edge-disjoint Hamiltonian 3n3/4 ln(n), then G contains at least α(δ,n)−3n2 (ln n) cycles. Proof. We follow Schema 18, turning Hamiltonian cycles of the host graph H into Hamiltonian cycles of the larger graph G.

Step 1: Choose p and q as given by Lemma 20. p and q are both within √ Hence, √ 1/4 1/2 1/4

2 2n + 4 of n , q is even, and a = n − pq ≤ 4 2n + 8. Note that the theorem is vacuously true for n < 160, 000. Hence we assume n ≥ 160, 000, and thus q is positive √ and a ≤ 10 q. Let P1 , . . . , Pq be a partition of the vertices of G given by Corollary 9. Let A be the set of parts that have size p + 1; note |A| = a. Let c be an integer such that q 3/4 ≤ c ≤ q 3/4 + 2 and b = q − a − c is odd. Let C be an arbitrary set of c parts disjoint from A. Let B contain all other parts; note |B| = b is odd. See Figure 7. Let H be the complete graph on vertex set A ∪ B ∪ C.

Step 2: Set j k p k = α(δ/q − (p + 1) ln(n + q), p) . √ Let τ = k − 9q ln q. Since we assumed n ≥ 160, 000 and using k ≤ p and the bounds on p and q in Lemma 20, we have that τ ≤q

21

(6)

(see Section 9.6 for details). Notice also that if n ≥ 160, 000, then q ≥ 367

(7)

(see Section 9.7 for details). Hence we can apply Lemma 27, obtaining a set F of Hamiltonian cycles of H satisfying the conditions of the lemma for the partition A, B, C of V (H) given in Step 1. In particular, |F| ≥ τ q/2 − 7q 7/4 .

Steps 3 and 4: We will find edge-disjoint matchings between parts, and assign these matchings to the Hamiltonian cycles. We combine Steps 3 and 4 of the schema because the edges within B need to be handled after we have blown up the other edges in H. By Corollary 9, the minimum degree of the induced bipartite subgraph between p every pair of parts is δ/q − (p + 1) ln(n + q). Note that p √ 3n3/4 ln n ≥ q (p + 1) ln(n + q) + q 3/2 (8) p for n ≥ 2 (see Section 9.8 for details), and hence δ is at least n/2+q (p + 1) ln(n + q)+ q 3/2 . Thus, the minimum degree of the induced bipartite subgraphs is at least p n/2 + q (p + 1) ln(n + q) + q 3/2 p √ − (p + 1) ln(n + q) ≥ p/2 + q. q Hence we can apply Csaba’s Theorem, Lemma 21, and Lemma 29. Fix an edge e of H with endpoints Pi and Pj . For every Hamiltonian cycle F ∈ F that uses edge e, we will associate a matching between Pi and Pj . The edges of the matching will be added to the subgraph F ∗ of G, which will eventually become a Hamiltonian cycle of G. The specifics of how to accomplish this are given in three steps.

Step I: e is within B or between B and C. In the induced bipartite subgraph between Pi and Pj , by Csaba’s Theorem there is a regular spanning subgraph of degree k (defined in Step√2), which decomposes into k perfect matchings. By Lemma 27, there are at most τ + 9q ln q = k cycles in F that use edge e. We arbitrarily assign edgedisjoint perfect matchings between Pi and Pj to the Hamiltonian cycles in F that contain e. Step II: e is between A and B. We assume pendpoint Pi is in A and Pj is in

B. By counting degrees, there are (p + 1)(δ/q − (p + 1) ln(n + q))/2 edges within Pi . Using bounds on p and q from Lemma 20 and n ≥ 160, 000, the number of edges within Pi is at least p (p + 1)(δ/q − (p + 1) ln(n + q))/2 ≥ δ/2 − n3/4 ln(n) (9)

(see Section 9.9 for details). We may assume that |F| ≤ δ/2−n3/4 ln(n) since otherwise we could remove Hamiltonian cycles from F as long as |F| ≥ α(δ, n)/2 − 5n7/8 ln(n). Thus there are at least as many edges within Pi as there are Hamiltonian cycles in F. For each Pi ∈ A, we create an injective map φPi from F to the edges within Pi . See Figure 8. For each F ∈ F that uses e, assume that F continues from Pi to some P` via

22

Figure 8: We route F through the part Pi of size p + 1 using an internal edge. the edge e0 . Arbitrarily assign the endpoints of φPi (F ), one to Pj and one to P` . For v ∈ Pi , let Φe,v Pi = {F ∈ F : e ∈ F and v is the endpoint of φPi (F ) assigned to Pj }. S e,v Note that t = v∈Pi ΦPi is at most k since there are at most k Hamiltonian cycles in F that contain e. The target f -factor for Lemma 21 applied to Pi and Pj is constant t on Pj and for v ∈ Pi , we set f (v) = t − Φe,v Pi . S Since | v∈Pi Φe,v Pi | = t ≤ k, the sum of f (v) over Pi and Pj are equal. Hence we can apply Lemma 21 to obtain matchings that saturate Pj , where |Φe,v Pi | of the matchings miss v ∈ Pi . Fix v ∈ Pi . Arbitrarily create a bijective assignment from the matchings between Pi and Pj that avoid v to the Hamiltonian cycles of Φe,v Pi .

Step III: e is within C. For each F ∈ F we define F ∗ to be the union of the matchings assigned to F in Steps I and II, along with {φPi (F ) : Pi ∈ A}. Note that F ∗ is a union of paths from Pi to Pj that span G. If we use an arbitrary perfect matching between Pi and Pj to expand e, the union of F ∗ and that matching will not necessarily be a Hamiltonian cycle of G, but instead be only a 2-factor, albeit a 2-factor where each cycle goes through each part. Instead, we choose a specific perfect matching that completes a Hamiltonian cycle. Let FPi ,Pj = {F ∈ F : F contains e}, enumerated as FPi ,Pj = {F1 , . . . , Ft }, and √ note t ≤ q by Lemma 27. We iteratively construct a perfect matching M` between Pi and Pj for each F` ∈ FPi ,Pj in the order ` = 1, . . . , t. Let T be the induced bipartite subgraph between Pi and Pj in G. We think of the paths in F`∗ as forming a perfect matching M`0 between Pi and Pj . Let T` be the bipartite subgraph with vertex set S 0 Pi ∪ Pj and edge set (E(T ) − `−1 T has minimum degree at i=1 Mi ) ∪ M` . The graph √ √ least p/2+ q, and hence T` has minimum degree p/2+ q−(`−1) ≥ p/2+1. Applying Lemma 29 to T` , we obtain a matching M` such that M` ∪ M`0 is a Hamiltonian cycle of T` . Iterating this procedure, we find edge-disjoint matchings {M` } for all F` ∈ FPi ,Pj . Thus, F`∗ ∪ M` is a Hamiltonian cycle of G. See Figure 9.

23

Figure 9: The union of paths in F`∗ can be extended to a Hamiltonian cycle by choosing a particular matching between Pi and Pj . Thus, we have |F| edge-disjoint Hamiltonian cycles of G. By Lemma 27, |F| ≥ √ 7/4 τ q/2 − 7q . Using τ = k − 9q ln q, |F| ≥

qk 3 3/2 p − q ln q − 7q 7/4 . 2 2

The variables δ, n, p, q, k are the same as in the proof of Theorem 23, and hence In√ equality 2 holds. Also using q ≤ n and n ≥ 160, 000, we have |F| ≥

α(δ, n) − 3n7/8 (ln n)1/4 . 2

(10)

See Section 9.10.

8

Future Work

Our results require the minimum degree of a graph to be beyond n/2 by an error term before finding edge-disjoint 1-factors or edge-disjoint Hamiltonian cycles. This seems to be an inherent consequence of our methods. If the minimum degree is exactly n/2 or very near n/2, what can be said? Question 30. Given a nonnegative constant c and a graph G on n vertices with n/2 ≤ δ(G) ≤ n/2 + c, how many edge-disjoint-1-factors does G contain? How many edge-disjoint Hamiltonian cycles does G contain? A lower bound is given by the theorem of Nash-Williams [21] stating that every such graph has b5n/224c edge-disjoint Hamiltonian cycles. An upper bound is given by Theorem 4 of Christofides, K¨ uhn, and Osthus [7].

References [1] Brian Alspach and Heather Gavlas. Cycle decompositions of Kn and Kn − I. J. Combin. Theory Ser. B, 81(1):77–99, 2001. [2] B. Bollob´ as and A. D. Scott. Problems and results on judicious partitions. Random Structures Algorithms, 21(3-4):414–430, 2002. Random structures and algorithms (Poznan, 2001).

24

[3] R. A. Brualdi. Probl`emes. In Probl`emes combinatoires et th´eorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 437–443. CNRS, Paris, 1978. [4] Arthur H. Busch, Michael J. Ferrara, Stephen G. Hartke, Michael S. Jacobson, Hemanshu Kaul, and Douglas B. West. Packing graphic n-tuples. In press, 2011. [5] Albert Bush. Two problems on bipartite graphs. Master’s thesis, Georgia State University, 2009. [6] Demetres Christofides, Daniela K¨ uhn, and Deryk Osthus. Hamilton decompositions of regular expanders: a proof of Kelly’s conjecture for large tournaments. To Appear in Advances in Math. [7] Demetres Christofides, Daniela K¨ uhn, and Deryk Osthus. Edge-disjoint Hamiltonian cycles in graphs. J. Comb. Theory, B, 102(5):1035–1060, 2012. [8] B´ela Csaba. Regular spanning subgraphs of bipartite graphs of high minimum degree. Electron. J. Combin., 14(1):Note 21, 7 pp. (electronic), 2007. [9] G.A. Dirac. Some theorems on abstract graphs. Proc. London Math. Soc. (3), 2:69–81, 1952. [10] Benjamin Doerr and Anand Srivastav. Multicolour discrepancies. Combin. Probab. Comput., 12(4):365–399, 2003. [11] Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, Cambridge, 2009. [12] Yoshimi Egawa. Edge-disjoint Hamiltonian cycles in graphs of Ore type. SUT J. Math., 29(1):15–50, 1993. [13] Alan Frieze and Michael Krivelevich. On packing hamilton cycles in -regular graphs. Journal of Combinatorial Theory, Series B, 94(1):159 – 172, 2005. [14] Alain Ghouila-Houri. Une condition suffisante d’existence d’un circuit hamiltonien. C. R. Acad. Sci. Paris, 251:495–497, 1960. [15] W. T. Gowers. Lower bounds of tower type for Szemer´edi’s uniformity lemma. Geom. Funct. Anal., 7(2):322–337, 1997. [16] K. J. Horadam. Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. [17] P. Katerinis. Minimum degree of a graph and the existence of k-factors. Proc. Indian Acad. Sci. Math. Sci., 94(2-3):123–127, 1985. [18] Daniela K¨ uhn and Deryk Osthus. Hamilton decompositions of regular expanders: applications. Preprint. [19] Sukhamay Kundu. The k-factor conjecture is true. Discrete Math., 6:367–376, 1973. [20] C. St. J. A. Nash-Williams. Hamiltonian lines in graphs whose vertices have sufficiently large valencies. In Combinatorial Theory and Its Applications, III (Proc. Colloq., Balatonf¨ ured, 1969), pages 813–819, Amsterdam, Netherlands, 1970. North-Holland.

25

[21] C. St. J. A. Nash-Williams. Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency. In Studies in Pure Mathematics (Presented to Richard Rado), pages 157–183. Academic Press, London, 1971. [22] Joel Spencer. Six standard deviations suffice. 289(2):679–706, 1985.

Trans. Amer. Math. Soc.,

[23] Douglas B. West. Introduction to graph theory. Prentice Hall Inc., Upper Saddle River, NJ, 1996.

9

Appendix: Details for Calculations

9.1

Inequality 1

Under the assumptions of the proof of Theorem 23, we show   p p (p + 1) δ/q − (p + 1) ln(n + q) ≥ δ − 2n3/4 ln(n). Using the bounds q ≤

√

n≤p≤

√

n+

√ √ 2 4 n + 4 from Lemma 20, we have

p p+1 δ − (p + 1)3 ln(n + q) q q √ √ √ √ ≥ δ − ( n + 2 4 n + 5)3 ln(n + n).

  p = (p + 1) δ/q − (p + 1) ln(n + q)

q p √ √ √ √ ( n + 2 4 n + 5)3 ln(n + n) ≤ 2n3/4 ln(n). To do so, let p us divide both sides by n3/4 ln(n). We need to show that q √ √ √ √ 1 √ 2≥ ( n + 2 4 n + 5)3 ln(n + n). n3/4 ln n

What is left to show is

We have 1 √

q √ √ √ √ ( n + 2 4 n + 5)3 ln(n + n)

n3/4 ln n v !3 u √ √ u 2 5 ln(n + n) t = 1+ √ +√ 4 ln(n) n n v !3 u √ u 2 5 ln(2n) ≤ t 1+ √ +√ 4 ln(n) n n v !3  u √  u 2 5 2 t ≤ 1+ √ +√ 1+ . 4 ln(n) n n

This expression is monotonically decreasing, and is less than 2 for n = 1000 by direct calculation, and hence will be less than 2 for n ≥ 1000.

26

9.2

Inequality 2

Under the assumptions of Theorem 23 and Theorem 5, we show Inequality 2: qk ≥ α(δ, n) − 1.42n7/8 (ln n)1/4 . First note that for n ≥ 64, that √ pq ≥ n − 4 2n1/4 − 8 √ √ ≥ n − 4 2n1/4 − 2 2n1/4 √ ≥ n − 6 2n1/4 . We now calculate p qk = q · α(δ/q − (p + 1) ln(n + q), p) q p p δ/q − (p + 1) ln(n + q) + 2(δ/q − (p + 1) ln(n + q))p − p2 =q (*) 2 q p p δ − (pqq + q 2 ) ln(n + q) + 2(δ − (pqq + q 2 ) ln(n + q))pq − (pq)2 = 2 q p p √ δ − (n3/2 + n)2 ln(n) + 2(δ − (n3/2 + n)2 ln(n))(n − 6 2n1/4 ) − n2 (**) ≥ 2 q p p √ δ − 2n3/4 ln(n) + 2(δ − 2n3/4 ln(n))(n − 6 2n1/4 ) − n2 ≥ 2 q p p √ √ 3/4 2 δ − 2n ln(n) + 2δn − n − 2(n − 6 2n1/4 )2n3/4 ln(n) − 12δ 2n1/4 ≥ q 2 p p √ √ √ 3/4 2 ln(n) + 2δn − n − 2(n − 6 2n1/4 )2n3/4 ln(n) + 12δ 2n1/4 δ − 2n ≥ 2 q p p √ 3/4 7/4 ln(n) + 4n ln(n) + 12 2δn1/4 2n ≥ α(δ, n) − 2 q p p √ 3/4 ≥ α(δ, n) − n ln(n) − n7/4 ln(n) + 3 2n5/4 because δ ≤ n. In step (*), under the large radical is a positive number because we called α(δ, n) with parameters where δ ≤ n. Later, in step (**) there may be a negative number under the √ √ √ radical, but when we apply x − y ≥ x − y, we will bound this imaginary number by a negative number, which is clearly less than the positive number we started with. W next bound the error term q p p √ n3/4 ln(n) + n7/4 ln(n) + 3 2n5/4 . We will do so with the term 1.42n7/8 (ln n)1/4 . Dividing by n7/8 (ln n)1/4 , we get the expression s √ (ln n)1/4 3 2 + 1+ √ . n1/8 n ln n This is monotonically decreasing, and less than 1.42 for n = 160, 000.

27

9.3

Inequality 3

We now bound

√ (8 2n1/4 + 14)k ≤ 1.57n7/8 (ln n)1/4 . √ √ √ Here, we will use k ≤ p ≤ n + 2 4 n + 5. Thus, we want a bound on √ √ √ √ (8 2n1/4 + 14)( n + 2 4 n + 5). If we divide this by n7/8 (ln n)1/4 , we obtain ! ! √ √ 8 2 14 2 5 + 1+ √ +√ . 4 n n n1/8 (ln n)1/4 n3/8 (ln n)1/4 This is clearly monotonically decreasing, and less than 1.57 for n = 160, 000.

9.4

Inequality 4

We now prove τ (b − 1)/2 ≥ τ q/2 − 0.53q 7/4 . √ For this we have that b ≥ q − 10 q − q 3/4 − 2 and τ ≤ q. Then we have 1 1 √ τ (b − 1) ≥ τ (q − 10 q − q 3/4 − 2) 2 2 √ ≥ τ q/2 − 5 qτ − q 3/4 τ /2 − τ ≥ τ q/2 − 5q 3/4 − q 7/4 /2 − q. Note that 5q 3/4 + q 7/4 /2 + q ≤ 0.53q 7/4 for q ≥ 367, and so the result follows.

9.5

Inequality 5

We now show

1 √ (q − q 3/4 )(10 q + q 3/4 + 2) < 1. 3 q

Showing the left hand side is monotone becomes easy by removing the factor of −q 3/4 . Then the expression we are dealing with simplifies to 1 √ (q)(10 q + q 3/4 + 2) ≤ 10/q 3/2 + 1/q 5/4 + 2/q 2 . 3 q This is strictly less than 1 for q ≥ 6.

9.6

Inequality 6

√ The next inequality is τ ≤ q. Since τ = k − 9q ln q and k ≤ p, we have p τ ≤ p − 9q ln q p √ √ ≤ n + 2n1/4 + 4 − 3 q ln q

28

√ 1/4 √ 1/4 √ √ We next show 3 q ln q ≥ 2 2n + 8. We will then have that n + √ 1/4 √2n 1/4 + 4 − √ √ √ ∗ 3 q ln q ≤ n− 2n√ −4, which√is bounded above by q. Set q = n− 2n √ −4; it is sufficient to show 3 q ∗ ln q ∗q≥ 2 2n1/4 +8. If we divide both sides by n1/4 ln q ∗ , we √ √ have the left side is equal to 3 1 − 2/n1/4 − 4/ n, which is monotonically increasing √ p √ in n, and the right side becomes 2 2/ ln(q ∗ )+8/(n1/4 ln q ∗ ), which is monotonically decreasing. Hence we just need to demonstrate the inquality for one value of n. If n = 500, then the left hand side is at least 80, and the right hand side is at most 25.

9.7

Inequality 7

√ √ We show that q ≥ 367 if n ≥ 160, 000. From Lemma 20, we have q > n − 2n1/4 − 4. √ √ For n ≥ 160, 000, n ≥ 20n1/4 , and hence q √ > (20 − 2)n1/4 − 4. This is clearly monotonically increasing, and hence q > (20 − 2)(160, 000)1/4 − 4 > 367.

9.8

Inequality 8

p √ √ We show 3n3/4 ln n ≥ q (p + 1) ln(n + q) + q 3/2 . Since pq ≤ n and q ≤ n, we have p p q (p + 1) ln(n + q) + q 3/2 = (pq · q + q 2 ) ln(n + q) + q 3/2 q ≤ (n3/2 + n)2 ln(n) + n3/4 p 4n3/2 ln n + n3/4 ≤ √ ≤ 3n3/4 ln n.

9.9

Inequality 9

We show

p √ (p + 1) ln(n + q))/2 ≥ δ/2 − n3/4 ln n. p By Inequality 1 in Section 9.1, we have (p + 1)(δ/q − (p + 1) ln(n + q) ≥ δ − p 2n3/4 ln(n), and the desired inequality follows simply by dividing both sides by 2. (p + 1)(δ/q −

9.10 We show

Inequality 10 α(δ, n) qk 3 3/2 p − q ln q − 0.53q 7/4 ≥ − 3/2n7/8 (ln n)1/4 . 2 2 2

Notice that by Inequality 2 in Section 9.2, qk ≥ α(δ, n) − 1.42n7/8 (ln n)1/4 . Hence we √ have, also using q ≤ n, qk 3 3/2 p α(δ, n) − q ln q − 0.53q 7/4 ≥ − 0.71n7/8 (ln n)1/4 − 2 2 2 α(δ, n) ≥ − 0.71n7/8 (ln n)1/4 − 2 α(δ, n) ≥ − 0.71n7/8 (ln n)1/4 − 2

29

3 3/2 p q ln q − 0.53q 7/4 2 q √ 3 3/4 n ln n − 0.53n7/8 2√ 3 2 3/4 √ n ln n − 0.53n7/8 . 4

Dividing the error term by n7/8 (ln n)1/4 . We have √ 3 2(ln n)1/4 + 0.53/(ln n)1/4 . 0.71 + 4n1/8 This is clearly monotonically decreasing and is less than 1.5 for n = 160, 000.

30