Lower Bounds and properties for the average number of colors in the non-equivalent colorings of a graph

TL;DR

This paper investigates the average number of colors in non-equivalent graph colorings, establishing properties, calculating values for specific graph classes, and proposing conjectures with proofs for certain graph types.

Contribution

It introduces new properties of the average coloring number, computes it for specific graphs, and proves conjectured bounds for classes like triangulated graphs.

Findings

Determined the average number of colors for certain graph classes.

Proved conjectured lower bounds for triangulated graphs.

Established properties of the invariant for various graphs.

Abstract

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several lower bounds on $\mathcal{A}(G)$ and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.