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 _xD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let _x(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, _xD(G) > _x(G) and _xD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on _xD(G) - _x(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 _xD(G) - _x(G) 6 1. In particular, we prove that _xD(K(n, 2)) = _x(K(n, 2)) + 1 for n > 5. In addition, we show that _xD(K(n, r)) = _x(K(n, r)) for n ≥ 2r + 1 and r ≥ 3.