Asymptotic multipartite version of the Alon–Yuster theorem

Abstract

In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemerédi theorem: If k ≥ 3 is an integer, H is a k-colorable graph and γ > 0 is fixed, then, for every sufficiently large n, where |V (H)| divides n, and for every balanced k-partite graph G on kn vertices with each of its corresponding k 2 bipartite subgraphs having minimum degree at least (k − 1)n/k + γn, G has a subgraph consisting of kn/|V (H)| vertex-disjoint copies of H.The proof uses the Regularity method together with linear programming.

An Asymptotic Multipartite Kühn--Osthus Theorem

“…In the case where gcd(H) = 1, the proof of Theorem 2 only uses the weaker assumption that h divides rn (rather than h divides n). However, as observed in [17], in the case gcd(H) > 1 we do indeed require that h divides n.…”

“…For each possible R, Lemma 9 would give us a perfect fractional (a ′ , b ′ )-weighted K r -tiling of R in which all weights are rational (see Remark 10). So, as observed in Section 3.2 from [17], there is a common denominator, bounded by a function of M , of all weights used in our perfect fractional (a ′ , b ′ )-weighted K r -tilings for each possible reduced graph R. Since 1/D ≪ 1/M , we may assume that D! is a multiple of this common denominator, and therefore that w(K)D!…”

“…We omit any proof of Proposition 14 in that it is a straightforward application of Szemerédi's Regularity Lemma and is implicit in many papers, including [2,14,17]. The degree form of Szemerédi's Regularity Lemma (see, for instance, [12, Theorem 1.10]) is sufficient here, modified for the multipartite setting.…”

“…However, for large n, Fischer's conjecture is 'almost-true', in that Catlin's counterexamples are the only counterexamples to the conjecture, as shown by Keevash and Mycroft [8] (this was previously demonstrated for r = 3 and r = 4 by Magyar and Martin [16] and Martin and Szemerédi [18] respectively, whilst an asymptotic form for all r was independently given by Lo and Markström [15]). Subsequently Martin and Skokan [17] continued this direction of research by establishing a multipartite analogue of the Alon-Yuster theorem, namely that for any H we have δ * (H, n) ≤ χ(H)−1 χ(H) n + o(n). Theorem 1 ( [17]).…”

“…Subsequently Martin and Skokan [17] continued this direction of research by establishing a multipartite analogue of the Alon-Yuster theorem, namely that for any H we have δ * (H, n) ≤ χ(H)−1 χ(H) n + o(n). Theorem 1 ( [17]). Let H be a graph on h vertices with χ(H) = r ≥ 3.…”