The minimum degree threshold for perfect graph packings

Abstract

Let H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show thatThe value of χ * (H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies χ(H) − 1 < χ * (H) ≤ χ(H).

Proof of the bandwidth conjecture of Bollobás and Komlós

“…For i ∈ [r − 1], we will fix colours l i , l i and positions s i and then obtain σ p+1 from σ p by a series of r − 1 appropriate (l i , l i , βn)-switches at positions s i . For this purpose recall that by induction (27) guarantees that σ p is (8rβn, 4rβn)-zero free on the interval [ pξ n, n]. When applied to the vertex t := pξ n + 12rβn, there exists a vertex t ∈ [pξ n + 12rβn, pξ n + 20rβn] such that [t , t +4rβn] is zero free.…”

“…Obviously, χ(H ) − 1 < χ cr (H ) ≤ χ(H ), with (approximate) equality for σ tending to 0 or |V (H )|/χ (H ), respectively. This concept was introduced by Komlós [21], who proved that a minimum degree condition of δ(G) ≥ (χ cr (H ) − 1)n/χ cr (H ) suffices to find a family of disjoint copies of H covering all but εn vertices of G. Kühn and Osthus [27] further investigated this question and managed to determine for every H the corresponding minimum degree condition (up to an additive constant) for the containment of a spanning H -factor.…”

Combinatorial and computational aspects of graph packing and graph decomposition

“…Namely, is it true that for each graph H, either χ(H) or χ cr (H) is the "right" constant to place in Theorem 2.9, guaranteeing an H-factor? Recently, Kühn and Osthus [59] gave a positive answer to this question. They prove that for any graph H, either its critical chromatic number or its chromatic number is the relevant parameter which guarantees the existence of H-factors in graphs of large minimum degree.…”

Note on Bipartite Graph Tilings

“…For a fixed graph H, necessary and sufficient conditions on the minimum-degree of G which guarantee that G contains an H-factor were studied extensively. Results in this spirit include the Tutte 1-factor Theorem (see [7]), the HajnalSzemerédi Theorem [4], and series of improving results by Alon and Yuster [1,2], Komlós [5], Zhao and Shokoufandeh [8], and by Kühn and Osthus [6]. In [6] the answer to the problem is settled (up to a constant) for any H. It was shown that the relevant parameters are the chromatic number and the critical chromatic number of H.…”