A characterization of graphs with small palette index

TL;DR

This paper characterizes graphs with small palette index, specifically those with index 2 or 3, by linking their structure to decompositions into regular subgraphs, providing a complete classification for regular graphs with index 3.

Contribution

It provides a complete characterization of graphs with palette index 2 or 3, connecting these properties to specific decompositions into regular subgraphs.

Findings

Graphs with palette index 1 are r-regular and r-edge-colorable.

Regular graphs with palette index 2 do not exist.

Complete characterization of regular graphs with palette index 3.

Abstract

Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of $G$. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index $1$ are $r$-regular graphs admitting an $r$-edge-coloring, while regular graphs with palette index $2$ do not exist. Here, we characterize all graphs with palette index either $2$ or $3$ in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index $3$.