Silver coloring of generalized Petersen graphs

In a graph G=(V,E), an independent set I(G) is a subset of the vertices of G such that no two vertices in I(G) are adjacent. Any maximum independent set of a graph is called a diagonal of the graph. Let c be a proper (r+1)-coloring of an r-regular graph G. A vertex v in G is said to be rainbow with respect to c if every color appears in the closed neighborhood N[v]=N(v)∪{v}. Given a diagonal I of G, the coloring c is said to be silver with respect to I if every v∈I is rainbow with respect to c. G is called silver if it admits a silver coloring with respect to some diagonal. In [1], the authors introduced silver coloring and the following question is raised “Find classes of r-regular graphs G, that G is a silver graph". This paper is aimed toward study this question for the generalized Petersen graphs. In this paper we show that, if n≡0 (mod4) and k is odd, then P(n,k) is a totally silver graph. Also, for every natural number n, the existence of silver coloring for generalized Petersen graphs P(n,1), P(n,2) except for n=5, this is well-known petersen graph, P(n,3) except for n=10,14 and 26. Also, for any k>2,P(2k+1,k), and for any k>3,P(3k+1,k), and for any k>3, k ≠5,9 P(3k-1,k) are silver graphs.