Factors of $r$-partite Graphs and Bounds for the Strong Chromatic Number

Abstract

We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order $3$. This result is used to give an upper bound of $2\mathrm{\Delta }$ for the strong chromatic number of $n$ vertex graphs with $\mathrm{\Delta }\ge n/6$.