Proof of a conjectured spectral upper bound on the chromatic number of a graph

TL;DR

This paper proves a spectral upper bound on the chromatic number of a graph, confirming a conjecture by Fan, Yu, and Wang, and extends previous bounds to the full range of chromatic numbers.

Contribution

It establishes a new spectral upper bound on the chromatic number involving the least eigenvalue, extending prior results and comparing with Wilf's bound.

Findings

Proved the conjectured spectral bound for all valid chromatic numbers.

Identified conditions for equality involving specific graph structures.

Compared the new bound with Wilf's bound and discussed their relative strengths.

Abstract

Let $G$ be a simple graph on $n$ vertices and $m$ edges with chromatic number $\chi$, and let $\lambda_n$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[Electron. J. Combin., 2012], we prove that when $3\le \chi\le n-1$, the chromatic number satisfies the following upper bound: $$ \chi \le \left(\frac{n}{2}+1+\lambda_n\right) + \sqrt{\left(\frac{n}{2}+1+\lambda_n\right)^{2}-4(\lambda_n+1)\left(\lambda_n+\frac{n}{2}\right)}, $$ with equality if and only if $G \cong \left(K_{\frac{\chi}{2}}\cup\tfrac{n-\chi}{2}K_1\right) \vee \left(K_{\frac{\chi}{2}}\cup\tfrac{n-\chi}{2}K_1\right)$, where both $n$ and $\chi$ are even. This extends the validity of the Fan--Yu--Wang bound from the range $3\le \chi\le \frac{n}{2}$ to the full range $3\le \chi\le n-1$. We also compare this bound with the well-known bound due to Wilf that $\chi \le 1 + \lambda_1$, where $\lambda_1$ denotes the largest eigenvalue. In particular we show that while Wilf's bound is an upper bound for some parameters larger than $\chi$, this bound using $\lambda_n$ is not an upper bound for these parameters. We conclude with a similar conjectured upper bound for $\chi(G)$, which uses $m$ in place of $n$.