Differential Coloring of Graphs

Abstract

Let $G=(V,E)$ be a connected graph. For a set $S\subseteq V$, the differential of $S$, denoted by $\partial (S)$, is defined to be $\partial (S)=\left|B(S)\right|-\left|S\right|$, where $B(S)=N(S)\cap (V-S)$. A set $S\subseteq V$ is said to be a positive differential set if $\partial (S)\ge 0$. A partition $\{V_1,V_2,\dots ,V_k\}$ of $V$ is said to be a positive differential chromatic partition of $G$ if each $V_i$ is both independent and positive differential. The minimum order of a positive differential chromatic partition of $G$ is called the differential chromatic number of $G$ and is denoted by ${\chi }_{\partial }(G)$. In this paper we initiate a study of this parameter.