Asymptotic multipartite version of the Alon-Yuster theorem

TL;DR

This paper proves an asymptotic multipartite extension of the Alon-Yuster theorem, establishing conditions under which a large balanced k-partite graph contains many disjoint copies of a k-colorable graph, using regularity and linear programming.

Contribution

It introduces an asymptotic multipartite version of the Alon-Yuster theorem, generalizing the Hajnal-Szemerédi theorem for large balanced k-partite graphs.

Findings

Establishes minimum degree conditions for multipartite graphs to contain disjoint copies of a k-colorable graph.

Uses regularity method and linear programming in the proof.

Generalizes classical theorems to the multipartite setting.

Abstract

In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi theorem: If $k\geq 3$ is an integer, $H$ is a $k$-colorable graph and $\gamma>0$ 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 $\binom{k}{2}$ bipartite subgraphs having minimum degree at least $(k-1)n/k+\gamma n$, $G$ has a subgraph consisting of $kn/|V(H)|$ vertex-disjoint copies of $H$. The proof uses the Regularity method together with linear programming.