An Asymptotic Multipartite Kühn--Osthus Theorem

Abstract

Abstract. In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph H with chromatic number r, the asymptotic multipartite minimum degree threshold which ensures that a large r-partite graph G admits a perfect H-tiling. We also give the threshold for an H-tiling covering all but a linear number of vertices of G, in a multipartite analogue of results of Komlós and of Shokoufandeh and Zhao.

Powers of Hamiltonian cycles in multipartite graphs

“…We will use the following lemma which can be found as a corollary to [27,Lemma 2.2]. (See also, [21,14].)…”

Triangle‐factors in pseudorandom graphs

“…Various other versions of the problem that put further restrictions on a graph, which, in turn, allow for a smaller sufficient minimum degree and therefore the minimum number of edges of graphs that satisfy it, have been studied. These include multipartite and Ramsey‐Túran versions . However, all these results require for some function depending on particular restrictions we put on a graph .…”

“…Various other versions of the problem that put further restrictions on a graph have been studied, with the common goal to reduce sufficient minimum degree and therefore the minimum number of edges of graphs that satisfy it. These include multipartite [11,14,15,21,22] and Ramsey-Túran versions [6,7]. However, all these results require δ(G) ≥ f (H)n for some function f depending on particular restrictions we put on a graph G. In particular, this does not tell us anything about graphs with o(n 2 ) edges.…”