Tomescu's graph coloring conjecture for $\ell$-connected graphs

TL;DR

This paper extends Tomescu's graph coloring conjecture to $ ext{ell}$-connected graphs, providing tight bounds on the number of proper colorings and characterizing extremal graphs for various connectivity levels.

Contribution

It establishes new bounds on proper colorings for $ ext{ell}$-connected graphs and characterizes extremal graphs, generalizing previous results to higher connectivity constraints.

Findings

Proves a tight bound for 2-connected graphs.

Provides an asymptotically tight bound for $ ext{ell} extgreater 2$.

Identifies unique extremal graphs for certain parameter ranges.

Abstract

Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asymptotically tight upper bound \[ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^n), \] and provide a matching lower bound construction. For the ranges $k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to connected graphs with minimum degree $\delta$.