On the edge r-dynamic chromatic number of some related graph operations

Novian Nur Fatihah (CGANT University of Jember Indonesia Mathematics Edu. Depart. University of Jember Indonesia)
Arika Indah Kriatiana (CGANT University of Jember Indonesia Mathematics Depart. University of Jember Indonesia)
Ika Hesti Agustin (CGANT University of Jember Indonesia Mathematics Edu. Depart. University of Jember Indonesia)
Dafik Dafik (CGANT University of Jember Indonesia Mathematics Depart. University of Jember Indonesia)

UNEJ e-Proceeding
2016: Proceeding The 1st International Basic Science Conference

All graphs in this paper are simple, nontrivial, connected and undirected. By an edge proper k-coloring of a graph G, we mean a map c : E(G) ! S, where jSj = k, such that any two adjacent edges receive different colors. An edge r-dynamic k-coloring is a proper k-coloring c of G such that jc(N(uv))j min (r; d(u) + d(v) ???? 2) for each edge uv in V (G), where N(uv) is the neighborhood of uv and c(S) = c(uv) : uv2S for an edge subset S. The edge r-dynamic chromatic number, written as r(G), is the minimum k such that G has an edge r-dynamic k-coloring. In this paper, we will determine the edge coloring r-dynamic number of a comb product of some graph, denote by G D H. Comb product of some graph is a graph formed by two graphs G and H, where each edge of graph G is replaced by which one edge of graph H.