Avoiding rainbow induced subgraphs in vertex-colorings

TL;DR

This paper characterizes when a graph necessarily contains a multicolored induced subgraph isomorphic to a fixed graph under any vertex coloring, showing that such unavoidable rainbow subgraphs are rare and explicitly identifying exceptions.

Contribution

It determines the induced vertex-anti-Ramsey number for all graphs and explicitly describes all cases where rainbow induced subgraphs are unavoidable.

Findings

Most graphs can avoid totally multicolored induced subgraphs isomorphic to a fixed graph.

The paper identifies all exceptional cases where rainbow subgraphs are unavoidable.

It fully characterizes when $G ightarrow H$ holds for given graphs $G$ and $H$.

Abstract

For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\rightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\rightarrow H$. We explicitly describe all exceptional cases. This determines the induced vertex-anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable.