On the local distinguishing chromatic number

Abstract

The distinguishing number of graphs is generalized in two directions by Cheng and Cowen (local distinguishing number) and Collins and Trenk (Distinguishing chromatic number). In this paper, we define and study the local distinguishing chromatic number of graphs. The exact values of 1-local distinguishing chromatic number of several classes of graphs such as stars, complete graphs, complete bipartite graphs, Petersen graph, cycles and paths are computed. Moreover, several bounds for local distinguishing chromatic number are presented.