On the Total Set Chromatic Number of Graphs

Abstract

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ s ( G ) of a graph G is the minimum number of colors required in a set coloring of G . In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = ( V, E ), its total graph T ( G ) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs.