# Characterization of forbidden subgraphs for bounded star chromatic number

## TL;DR

This paper characterizes which forbidden subgraphs ensure bounded star and acyclic chromatic numbers in graph classes, extending understanding of coloring parameters that avoid specific bicolored subgraphs.

## Contribution

It provides a complete characterization of graphs F where classes with no F as a subgraph have bounded H-avoiding chromatic numbers, especially for star and acyclic colorings.

## Key findings

- Characterization of graphs F for bounded star chromatic number
- Complete characterization for acyclic chromatic number
- Extension of coloring parameter bounds to classes excluding certain subgraphs

## Abstract

The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Gr\"unbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbids a bicolored subgraph $H$. When considering the class $\mathcal C$ of graphs with no $F$ as an induced subgraph, it is not hard to see that every graph in $\mathcal C$ has bounded chromatic number if and only if $F$ is a complete graph of size at most two. We study this phenomena for the class of graphs with no $F$ as a subgraph for $H$-avoiding coloring. We completely characterize all graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded $H$-avoiding chromatic number for a large class of graphs $H$. As a corollary, our main result implies a characterization of graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.