Upper bounds on the average number of colors in the non-equivalent colorings of a graph

TL;DR

This paper establishes upper bounds on the average number of colors in non-equivalent colorings of a graph, providing general and specific bounds for certain graph classes.

Contribution

It introduces a universal upper bound on the average number of colors in non-equivalent colorings and refines bounds for graphs with specific degrees.

Findings

Derived a general upper bound valid for all graphs.

Provided a more precise bound for graphs with maximum degree 1, 2, or n-2.

Enhanced understanding of coloring complexity in various graph classes.

Abstract

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be the average number of colors in the non-equivalent colorings of a graph $G$. We give a general upper bound on $\mathcal{A}(G)$ that is valid for all graphs $G$ and a more precise one for graphs $G$ of order $n$ and maximum degree $\Delta(G)\in \{1,2,n-2\}$.