An asymptotic multipartite K\"uhn-Osthus theorem

TL;DR

This paper establishes asymptotic minimum degree thresholds in large multipartite graphs that guarantee perfect or near-perfect tilings with a fixed graph H, extending classical theorems to the multipartite setting.

Contribution

It proves an asymptotic multipartite version of the K"uhn-Osthus theorem, providing thresholds for perfect and almost-perfect H-tilings in large multipartite graphs.

Findings

Determined the minimum degree threshold for perfect H-tilings in multipartite graphs.

Established thresholds for near-complete H-tilings covering all but a linear number of vertices.

Extended classical tiling results to the multipartite graph setting.

Abstract

In this paper we prove an asymptotic multipartite version of a well-known theorem of K\"uhn 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 Koml\'os and of Shokoufandeh and Zhao.