The neighbor coloring set in graphs
B. Chaluvaraju, C.N. Prasanna Kumar, C. Appajigowda · International journal of applied mathematics and computation · 2012
Given a graph $G=(V, E)$, a set $S\subseteq V$ is a neighborhood set of $G$, if $G = \bigcup_{v\in S}\langle N[v]\rangle$, where $\langle N[v]\rangle$ is the sub graph of $G$ induced by $v$ and all vertices adjacent to $v$. A neighborhood set $S\subseteq V$ is said to be a neighbor coloring set of $G$ if each color class $V_{i}, 1\leq i \leq k$ contains at least one vertex, which belongs to $S$. The minimum cardinality taken over all neighbor coloring set of a graph $G$ is called neighbor chromatic number and is denoted by $\chi_{\eta}(G)$. In this paper, we study the properties of $\chi_{\eta}(G)$ and also its relationship with other graph theoretic parameters are explored.