Coloring graphs with dense neighborhoods

TL;DR

This paper establishes new bounds on the chromatic number of graphs based on maximum degree, local average degree, and clique size, providing improved coloring criteria and structural insights.

Contribution

It introduces novel bounds relating local neighborhood density to global coloring properties, extending classical graph coloring results.

Findings

Graphs with high local neighborhood density are either nearly optimally colorable or contain large cliques.

Improved bounds for the case k=1 on average degree and clique size.

New inequalities linking chromatic number, clique number, maximum degree, and independence number.

Abstract

It is shown that any graph with maximum degree $\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}\Delta + k + 6$ is either $(\Delta - k)$-colorable or contains a clique on more than $\Delta - 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\frac23\Delta + 4$ and the bound on the clique number to $\Delta-1$. As corollaries, we show that every graph satisfies $\chi \leq \max\set{\omega, \Delta - 1, 4\alpha}$ and every graph satisfies $\chi \leq \max\set{\omega, \Delta - 1, \ceil{\frac{15 + \sqrt{48n + 73}}{4}}}$.