The Distinguishing Chromatic Number of Kneser Graphs

A labeling f : V (G) → {1, 2, . . . , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let χ(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, χD(G) > χ(G) and χD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on χD(G) − χ(G) in terms of the order of the automorphism group of G, improved when the automorphism group of G is a finite abelian group. The Kneser graph K(n, r) is a graph whose vertices are the r-subsets of an n element set, and two vertices of K(n, r) are adjacent if their corresponding two r-subsets are disjoint. In this paper, we provide a class of graphs G, namely Kneser graphs K(n, r), whose automorphism group is the symmetric group, Sn, such that χD(G)− χ(G) 6 1. In particular, we prove that χD(K(n, 2)) = χ(K(n, 2)) + 1 for n > 5. In addition, we show that χD(K(n, r)) = χ(K(n, r)) for n > 2r + 1 and r > 3.