Non-degenerate colorings in the Brook's Theorem

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.