On the maximum number of colorings of a graph

TL;DR

This paper proves a conjecture about the maximum number of colorings in certain connected graphs with a restriction on independence number, extending known results for specific chromatic numbers.

Contribution

It establishes the maximum number of colorings for connected $k$-chromatic graphs with independence number at most 2, confirming a conjecture for this class.

Findings

Proves the conjecture for graphs with independence number ≤ 2.

Identifies the extremal graphs achieving maximum colorings.

Extends understanding of graph coloring bounds for specific graph classes.

Abstract

Let $\mathcal{C}_k(n)$ be the family of all connected $k$-chromatic graphs of order $n$. Given a natural number $x\geq k$, we consider the problem of finding the maximum number of $x$-colorings among graphs in $\mathcal{C}_k(n)$. When $k\leq 3$ the answer to this problem is known, and when $k\geq 4$ the problem is wide open. For $k\geq 4$ it was conjectured that the maximum number of $x$-colorings is $x(x-1)\cdots (x-k+1)\,x^{n-k}$. In this article, we prove this conjecture under the additional condition that the independence number of the graphs is at most $2$.