A degree sequence Hajnal–Szemerédi theorem

Abstract

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal-Szemerédi theorem [12] characterises the minimum degree that ensures a graph G contains a perfect Kr-packing. Balogh, Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal-Szemerédi theorem which, if true, gives a strengthening of the Hajnal-Szemerédi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon-Yuster theorem [3] which gives a minimum degree condition that ensures a graph contains a perfect H-packing for an arbitrary graph H. We give a wide-reaching generalisation of this result by answering another conjecture of Balogh, Kostochka and Treglown [4] on the degree sequence of a graph that forces a perfect H-packing. We also prove a degree sequence result concerning perfect transitive tournament packings in directed graphs. The proofs blend together the regularity and absorbing methods.Date: September 22, 2018.1 Conjecture 1.2 (Balogh, Kostochka and Treglown [4]). Let n, r ∈ N such that r divides n. Suppose that G is a graph on n vertices with degree sequence d 1 ≤ · · · ≤ d n such that:(α) d i ≥ (r − 2)n/r + i for all i < n/r; (β) d n/r+1 ≥ (r − 1)n/r. Then G contains a perfect K r -packing.Note that Conjecture 1.2, if true, is much stronger than the Hajnal-Szemerédi theorem since the degree condition allows for n/r vertices to have degree less than (r − 1)n/r. Moreover, the degree sequence condition in Conjecture 1.2 is best-possible. Indeed, examples in Section 4 in [4] show that one cannot replace (α) with d i ≥ (r − 2)n/r + i − 1 for a single value of i < n/r and (β) cannot be replaced with d n/r+1 ≥ (r − 1)n/r − 1. Chvátal's theorem on Hamilton cycles [6] implies Conjecture 1.2 in the case when r = 2. In [4] it was shown that Conjecture 1.2 is true under the additional assumption that no vertex x ∈ V (G) of degree less than (r − 1)n/r lies in a copy of K r+1 .In this paper we prove the following asymptotic version of Conjecture 1.2.Theorem 1.3. Let γ > 0 and r ∈ N. Then there exists an integer n 0 = n 0 (γ, r) such that the following holds. Suppose that G is a graph on n ≥ n 0 vertices where r divides n and with degree sequence d 1 ≤ · · · ≤ d n such that:Keevash and Knox [14] also have announced a proof of Theorem 1.3 in the case when r = 3. Conjecture 1.2 considers a 'Pósa-type' degree sequence condition: Pósa's theorem [25] states that a graph G on n ≥ 3 vertices has a Hamilton cycle if its degree sequence d 1 ≤ · · · ≤ d n satisfies d i ≥ i+1 for all i < (n−1)/2 and if additionally d ⌈n/2⌉ ≥ ⌈n/2⌉ when n is odd. The aforementioned result of Chvátal [6] generalises Pósa's theorem by characterising those degree sequences which force Hamiltonicity. It would be interesting to establish an analogous result for perfect K r -packings.After this paper was submitted, the author and Staden [27] have used Theorem 1.3 to prove a degree sequence version of Pósa's conjecture on ...

On Degree Sequences Forcing The Square of a Hamilton Cycle

“…Suppose that G is a sufficiently large graph on n vertices as in the statement of Theorem 1.3. A result of the second author [40] guarantees that G contains a collection of n/3 vertex-disjoint triangles (see Theorem 5.2). Further, this result together with a simple application of the Regularity lemma implies that G in fact contains a collection P of a bounded number of vertex-disjoint square paths that together cover almost all of the vertices in G. So we can indeed prove an analogue of Step 3 in this setting.…”

“…To prove Lemma 5.1, we will use the following result of the second author [40] which guarantees a perfect triangle packing in a sufficiently large η-good graph.…”

“…Theorem 5.2. [40] For every η > 0, there exists n 0 ∈ N such that every η-good graph G on n ≥ n 0 vertices contains a perfect K 3 -packing.…”

A Degree Sequence Komlós Theorem

“…The proof of the next result is analogous to that of Lemma 5.2 in [16]. It states that the degree sequence of G in Theorem 4.1 is 'inherited' by its reduced graph R.…”

“…We first show that it suffices to prove Theorem 4.1 in the case when H = B, a bottle graph with neck σ and width ω (where σ < ω). In particular, Theorem 4.1 is already known in the case when H is a balanced r-partite graph [16].…”