Tiling tripartite graphs with 3-colorable graphs: The extreme case

TL;DR

This paper establishes a precise minimum degree condition for perfectly tiling large tripartite graphs with copies of a complete tripartite graph, extending previous results and identifying tight bounds.

Contribution

It provides a new extremal condition for tiling tripartite graphs with 3-colorable graphs, generalizing earlier work and determining tight bounds for the minimum degree requirement.

Findings

Established a minimum degree threshold for perfect tiling with $K_{h,h,h}$.

Extended previous results to more general 3-colorable graphs.

Proved the tightness of the bounds under certain divisibility conditions.

Abstract

There is a sufficiently large $N\in h\mathbb{N}$ such that the following holds. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $2N/3+2h-1$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends work by two of the authors [Electron. J. Combin, 16(1), 2009] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that $2N/3+2h-1$ in our result can not be replaced by $2N/3+ h-2$ and that if $N$ is divisible by $6h$, then we can replace it with the value $2N/3+h-1$ and this is tight.