Monochromatic Connectivity in Monochromatic-star-free Graphs

Abstract

We consider edge-colorings of complete graphs in which each color induces a subgraph that does not contain an induced copy of $K_{1,t}$, for some $t\ge 3$. It turns out that such colorings, if the underlying graph is sufficiently large, contain spanning monochromatic $k$-connected subgraphs. Furthermore, there exists a color, say blue, such that every vertex has very few incident edges in colors other than blue.