Extremal Values of the Chromatic Number for a Given Degree Sequence

TL;DR

This paper investigates the extremal values of the chromatic number for graphs with a fixed degree sequence, providing bounds and polynomial-time computability results for these parameters.

Contribution

It establishes new bounds for the minimum and maximum chromatic numbers given a degree sequence and shows these can be computed efficiently.

Findings

Bounds for _{\,min}(d) and _{\,max}(d) based on degree sequence properties

Exact formula for _{\,max}(d) under certain conditions

Polynomial-time algorithms for computing _{\,min}(d), _{\,max}(d), and _{\,min}(d)

Abstract

For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $\chi_{\min}(d)$ and the largest chromatic number $\chi_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then $\chi_{\min}(d)\leq \max\left\{ 3,d_1-\frac{n+1}{4d_1}+4\right\}$, and, if $\sqrt{n+\frac{1}{4}}-\frac{1}{2}>d_1\geq d_n\geq 1$, then $\chi_{\max}(d)=\max\limits_{i\in [n]}\min\left\{ i,d_i+1\right\}$. For a given degree sequence $d$ with bounded entries, we show that $\chi_{\min}(d)$, $\chi_{\max}(d)$, and also the smallest independence number $\alpha_{\min}(d)$ among all graphs with degree sequence $d$, can be determined in polynomial time.