The list distinguishing number of Kneser graphs

Abstract

A graph G is said to be k-distinguishable if the vertex set can be colored using k colors such that no non-trivial automorphism fixes every color class, and the distinguishing number D(G) is the least integer k for which G is k-distinguishable. If for each v ∈ V (G) we have a list L(v) of colors, and we stipulate that the color assigned to vertex v comes from its list L(v) then G is said to be L-distinguishable where L = {L(v)} v∈V (G) . The list distinguishing number of a graph, denoted D l (G), is the minimum integer k such that every collection of lists L with |L(v)| = k admits an L-distinguishing coloring. In this paper, we prove that D l (G) = D(G) when G is a Kneser graph.An interesting variant of the distinguishing number of a graph, due to Ferrara, Flesch, and Gethner [6] goes as follows. Given an assignment L = {L(v)} v∈V (G) of lists of available colors to vertices of G, we say that G is L−distinguishable if there is a distinguishing coloring f of G such that f (v) ∈ L(v) for all v. The list distinguishing number of G, D l (G) is the minimum integer k such that G is L−distinguishable for any list assignment L with |L(v)| = k for all v. The list distinguishing number has generated a bit of interest recently (see [6,7,8] for some relevant results) primarily due to the following conjecture that appears in [6]: For any graph G, D l (G) = D(G). The paper [6], in which this notion was introduced and the conjecture was made, proves the same for cycles of size at least 6, cartesian products of cycle, and for graphs whose automorphism group is the Dihedral group. The paper [7] proves the validity of the conjecture for trees, and [8] establishes it for interval graphs.Let r ≥ 2, and n ≥ 2r + 1. The Kneser graph K(n, r) is defined as follows: The vertex set of K(n, r) consists of all k−element subsets of [n]; vertices u, v in K(n, r) are adjacent if and only if u ∩ v = ∅. The Distinguishing number of the Kneser graphs is well known (see [2]): D(K(n, r)) = 2 when n = 2r + 1 and r ≥ 3; for r = 2, D(K(5, 2)) = 3, and D(K(n, 2)) = 2 for all n ≥ 6.Our main theorem in this short paper establishes the validity of the list distinguishing conjecture for the family of Kneser graphs.