Some results on the palette index of graphs

TL;DR

This paper investigates the palette index of graphs, providing bounds and conjectures, especially for bipartite graphs with small degrees, and characterizes graphs with maximum palette index.

Contribution

It introduces new bounds on the palette index based on vertex degrees and proves the conjecture for certain bipartite graph families.

Findings

Bounds on palette index in terms of vertex degrees

Conjecture that palette index for bipartite graphs is at most 1 plus the maximum degree

Characterization of graphs with palette index equal to the number of vertices

Abstract

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices.