Minimum degree thresholds for bipartite graph tiling

Abstract

For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we prove a corresponding minimum degree threshold to guarantee that $G$ has an $H$-tiling missing only a constant number of vertices. Our threshold for the perfect tiling depends on either the chromatic number $\chi(H)$ or the critical chromatic number $\chi_{cr}(H)$ while the threshold for the almost perfect tiling only depends on $\chi_{cr}(H)$. Our results answer two questions of Zhao. They can be viewed as bipartite analogs to the results of Kuhn and Osthus and of Shokoufandeh and Zhao

An Asymptotic Multipartite Kühn--Osthus Theorem

“…He conjectured that in fact this condition guarantees an H-tiling covering all but a constant number of vertices, and this was subsequently confirmed by Shokoufandeh and Zhao [22]. Our second main result (Theorem 5) gives a multipartite analogue of Komlós's result, namely that any χ(H)-partite graph G whose vertex classes each have size n and which satisfies δ * (G) ≥ χcr(H)−1 χcr(H) n admits an H-tiling covering all but o(n) vertices of G. Again, an analogous result for bipartite graphs H was previously given by Bush and Zhao [2].…”

“…Constructions given in Section 5 demonstrate that, for any graph H, Theorem 2 is best-possible up to the αn error term in the degree condition. A similar but slightlydifferent result holds in the case r = 2; this case was fully settled by Bush and Zhao [2] up to an additive constant.…”

“…Theorem 4 (Bush-Zhao [2]). For any bipartite graph H there exist constants n 0 = n 0 (H) and c = c(H) such that if G is a balanced bipartite graph with n ≥ n 0 vertices in each part, and |H| divides 2n, then the following statements hold (where gcd(H) is defined as above and gcd cc (H) is the greatest common divisor of the sizes of the connected components of H).…”

“…Theorem 6 (Bush-Zhao [2]). For any bipartite graph H, there exist constants n 0 = n 0 (H) and c = c(H) such that whenever G is a balanced bipartite graph with n ≥ n 0 vertices in each part, δ(G) ≥ (1 − 1/χ cr (H))n suffices to ensure an H-tiling of G that covers all but at most c vertices.…”

“…In this paper we prove an asymptotic multipartite analogue of the Kühn-Osthus theorem (Theorem 2), which establishes the asymptotic value of δ * (H, n) for any graph H with χ(H) ≥ 3. Together with a theorem of Bush and Zhao [2], who previously gave the corresponding result for bipartite graphs H up to an additive constant, this determines the asymptotic value of δ * (H, n) for every graph H.…”