Tiling tripartite graphs with 3-colorable graphs

TL;DR

This paper establishes conditions under which large tripartite graphs can be perfectly tiled with copies of K_{h,h,h}, extending previous results and providing tight bounds for tiling with 3-colorable graphs.

Contribution

It proves a new minimum-degree condition for tiling tripartite graphs with K_{h,h,h} and extends to tiling with any fixed 3-colorable graph, with tight bounds.

Findings

Tiling is possible under the specified minimum-degree condition for large N.

The minimum-degree condition is proven to be best possible.

Provides tight bounds for cases where N is divisible by h but not by 6h.

Abstract

For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently large, then G can be tiled perfectly by copies of K_{h,h,h}. This extends the work in [19] and also gives a sufficient condition for tiling by any (fixed) 3-colorable graph. Furthermore, we show that this minimum-degree condition is best possible and provide very tight bounds when N is divisible by h but not by 6h.