Tiling Tripartite Graphs with 3-Colorable Graphs: The Extreme Case

Abstract

There is a sufficiently large N ∈ hN 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.