Minimum degree thresholds for bipartite graph tiling

TL;DR

This paper establishes optimal minimum degree thresholds for bipartite graphs to contain perfect or near-perfect tilings with a fixed bipartite graph H, extending previous results and answering open questions.

Contribution

It determines the exact minimum degree thresholds for perfect and almost perfect H-tilings in bipartite graphs, depending on chromatic parameters, improving and generalizing prior work.

Findings

Thresholds depend on $ ext{chi}(H)$ or $ ext{chi}_{cr}(H)$ for perfect tilings.

Threshold for almost perfect tilings depends only on $ ext{chi}_{cr}(H)$.

Results are tight up to a constant factor depending on H.

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.