Non-degenerate colorings in the Brook's Theorem

TL;DR

This paper introduces a new type of graph coloring called $(c,p)$-nondegenerate coloring, generalizing Brook's Theorem by ensuring vertices with high degree have diverse neighboring colors, with proofs and corollaries for graphs with bounded degree and clique restrictions.

Contribution

It extends Brook's Theorem to $(c,p)$-nondegenerate colorings, providing existence results for graphs with degree constraints and no large cliques.

Findings

Existence of $(c,p)$-nondegenerate colorings for graphs without large cliques.

Generalization of Brook's Theorem to new coloring conditions.

Derivation of interesting corollaries from the primary proof.

Abstract

Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let $D\geq 3$ and $G$ be a graph without cliques on $D+1$ vertices and the degree of any vertex in this graph is not greater than $D$. Then for every integer $c\geq 2$ there is a proper $(c,p)$-nondegenerate vertex $D$-coloring of $G$, where $p=(c^3+8c^2+19c+6)(c+1).$ During the primary proof, some interesting corollaries are derived.