On the locating chromatic number of Kneser graphs

Abstract

Let c be a proper k -coloring of a connected graph G and Π = ( C 1 , C 2 , … , C k ) 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 , C 1 ) , d ( v , C 2 ) , … , d ( v , C k ) ) , where d ( v , C i ) = min { d ( v , x ) | x ∈ C i } , 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 χ L ( G ) . In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ L ( K G ( n , 2 ) ) = n − 1 for all n ≥ 5 . Then, we prove that χ L ( K G ( n , k ) ) ≤ n − 1 , when n ≥ k 2 . Moreover, we present some bounds for the locating chromatic number of odd graphs.