Proof of a conjectured spectral upper bound on the chromatic number of a graph
Quanyu Tang, Clive Elphick
TL;DR
The paper settles a conjecture on a spectral upper bound for the chromatic number by proving an explicit inequality linking χ to the least adjacency eigenvalue λn for all 3≤χ≤n−1. It replaces the original calculus-based approach with a purely algebraic reduction to a symmetric extremal graph G(χ/2,(n−χ)/2), derives the lower bound for λn and characterizes equality, and thereby yields a closed-form χ-bound in terms of n and λn. A detailed comparison with Wilf's bound clarifies regimes where the new bound is or is not superior, and examples show their incomparability in general. The work confirms the conjecture and highlights parity-dependent complications that remain for a completely uniform proof.
Abstract
Let $G$ be a simple graph on $n$ vertices with chromatic number $χ$, and let $λ_n$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[Electron. J. Combin., 2012], we prove that when $3\le χ\le n-1$, the chromatic number satisfies the following upper bound: $$ χ\le \left(\frac{n}{2}+1+λ_n\right) + \sqrt{\left(\frac{n}{2}+1+λ_n\right)^{2}-4(λ_n+1)\left(λ_n+\frac{n}{2}\right)}, $$ with equality if and only if $G \cong \left(K_{\fracχ{2}}\cup\tfrac{n-χ}{2}K_1\right) \vee \left(K_{\fracχ{2}}\cup\tfrac{n-χ}{2}K_1\right)$, where both $n$ and $χ$ are even. This extends the validity of the Fan--Yu--Wang bound from the range $3\le χ\le \frac{n}{2}$ to the full range $3\le χ\le n-1$. We also compare this bound with the well known bound due to Wilf that $χ\le 1 + λ_1$, where $λ_1$ denotes the largest eigenvalue. In particular we show that while Wilf's bound is an upper bound for some parameters larger than $χ$, this bound using $λ_n$ is not an upper bound for these parameters.