A Note on General Neighbor-Distinguishing Total Coloring of Graphs

Abstract

The general neighbor-distinguishing total chromatic number \(\chi”_{gnd}(G)\) of a graph \(G\) is the smallest integer \(k\) such that the vertices and edges of \(G\) can be colored by \(k\) colors so that no adjacent vertices have the same set of colors. It is proved in this note that \(\chi”_{gnd}(G) = \lceil \log_2 \chi(G) \rceil + 1\), where \(\chi(G)\) is the vertex chromatic number of \(G\).