A note on bipartite graph tiling

TL;DR

This paper refines minimum degree conditions for bipartite graph tiling, particularly addressing the case when the number of copies is odd and the bipartite subgraph size t exceeds 2s+1, completing previous results.

Contribution

It establishes the best possible minimum degree condition for bipartite graph tiling when m is odd and t>2s+1, filling a gap in prior research.

Findings

Provides optimal degree condition for m odd, t>2s+1 case.

Completes the characterization of bipartite tiling conditions.

Builds on and extends previous results by Zhao and Hladký and Schacht.

Abstract

Bipartite graph tiling was studied by Zhao who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s}. Let s<t be fixed positive integers. Hladk\'y and Schacht gave minimum degree conditions for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t}. Their results were best possible, except in the case when m is odd and t> 2s+1. We give the best possible minimum degree condition in this case.