The Distinguishing Chromatic Number

Abstract

In this paper we define and study the distinguishing chromatic number, $\chi_D(G)$, of a graph $G$, building on the work of Albertson and Collins who studied the distinguishing number. We find $\chi_D(G)$ for various families of graphs and characterize those graphs with $\chi_D(G)$ $ = |V(G)|$, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks' Theorem for both the distinguishing number and the distinguishing chromatic number, and for both trees and connected graphs. We conclude with some conjectures.

Base size, metric dimension and other invariants of groups and graphs

“…The partition is commonly regarded as a labelling or colouring of the vertex set. This colouring is not necessarily a proper graph colouring (where adjacent vertices have different colours), but this extra assumption has been considered by Collins and Trenk [41]; the number of colours needed is referred to as the distinguishing chromatic number of the graph.…”

Distinguishing number and distinguishing index of certain graphs

“…The distinguishing index of some examples of graphs was exhibited in [8]. In the sequel, we need the following results: Theorem 1.1 [3,9] If G is a connected graph with maximum degree ∆, then D(G) ≤ ∆ + 1. Furthermore, equality holds if and only if G is a K n , K n,n , C 3 , C 4 or C 5 .…”

On the Distinguishing Number of Functigraphs

“…Harary [18] gave different methods (orienting some of the edges, coloring some of the vertices with one or more colors and same for the edges, labeling vertices or edges, adding or deleting vertices or edges) of destroying the symmetries of a graph. Collins and Trenk defined the distinguishing chromatic number in [13] where they used proper t-distinguishing for vertex labeling. They have also given a comparison between the distinguishing number, the distinguishing chromatic number and the chromatic number for families like complete graphs, paths, cycles, Petersen graph and trees etc.…”