Distinguishing Chromatic Number of Cartesian Products of Graphs

Abstract

The distinguishing chromatic number χ D (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(•). Our main result is that for every graph G, there exists a constant d G such that for all d ≥ d G the distinguishing chromatic number of G d is at most χ(G)+ 1, where G d is the Cartesian product of d copies of G. We also prove that for d ≥ 5, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.

χ_D(G), |Aut(G)|, and a variant of the motion lemma

“…As a contrast, we also demonstrate a family of graphs with arbitrarily large chromatic number, with 'super large' automorphism groups for which every proper coloring of G with χ(G) colors is in fact distinguishing. This latter example also addresses a point raised in [3] and these contrasting results indicate that the relation between |Aut(G)| and χ D (G) can tend to be haphazard.…”

“…The distinguishing chromatic number of a Cartesian product of graphs has been studied in [3]. The fact that any graph can be uniquely (upto a permutation of the factors) factorized into prime graphs with respect to the Cartesian product plays a pivotal role in determining the full automorphism group.…”

“…The problem of determining the distinguishing chromatic number of a graph G, or at least good bounds for it, has been one of considerable interest in recent times ( [4,13,5,2,3]). Clearly, the notion of the distinguishing chromatic number begins to get more interesting only if the graph admits a large group of automorphisms, in which case, it can vary substantially from the usual chromatic number.…”

“…Some upper bounds for χ D (G) (for instance, a version of Brooks' theorem for the distinguishing chromatic number) appear in [4], which also includes the inequality χ D (G) ≤ D(G)χ(G). However, in many interesting large naturally occurring families of graphs, we have χ D (G) ≤ χ(G) + 1 (see [3,2,5,4]). …”

Improving upper bounds for the distinguishing index

“…Both numbers, D(G) and χ D (G), have been intensively investigated by many authors in recent years [4,5,6,9,16]. Our investigation was motivated by the renowned result of Nordhaus-Gaddum [18] who proved in 1956 the following lower and upper bounds for the sum of the chromatic numbers of a graph and its complement (actually, the upper bound was first proved by Zykov [22] in 1949).…”