On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G andΠ = (C1, C2, . . . , Ck) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c Π (v) := (d(v, C1), d(v, C2), . . . , d(v, Ck)), where d(v, Ci) = min{d(v, x)|x ∈ Ci}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χL (G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χL (KG(n, 2)) = n − 1 for all n ≥ 5. Then, we prove that χL (KG(n, k)) ≤ n − 1, when n ≥ k2. Moreover, we present some bounds for the locating chromatic number of odd graphs. © 2011 Elsevier B.V. All rights reserved.