INJECTIVE COLORING OF GENERALIZED PETERSEN GRAPHS

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. The injective chromatic number xi(G) of a graph G is the smallest number k such that G admits an injective coloring with k colors. Hahn et al.(2002) proved that Delta <= chi(i)(G) <= Delta(2)- Delta + 1 for any graph G, where Delta is the maximum degree of G. For a constant c >= 0, determining the injective chromatic number of which graphs is at most Delta +c is an interesting problem. In this paper, we investigate the injective colorings of generalized Petersen graphs P(n,k). We prove that chi(i)(P(m, k)) <= 5 for any generalized Petersen graph P(n, k) and chi(i)(P(n,k)) = 3 if n 0 (mod 3) and k not equivalent to 0 (mod 3). Furthermore, we determine the precise injective chromatic numbers of P(n, 1) and P(n, 2).