On The Total r-dynamic Coloring of Edge Comb Product graph G D H

Dwi Agustin Retno Wardani (Mathematics Depart. University of Jember)
Dafik Dafik (CGANT University of Jember Mathematics Edu. Depart. University of Jember)
Antonius C. Prihandoko (CGANT University of Jember System Information. Depart. University of Jember)
Arika I. Kristiana (CGANT University of Jember Mathematics Edu. Depart. University of Jember)

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

Given that two natural numbers r, k. By a proper total k-coloring of a graph G, we mean a map c : V (G) ∪ E(G) → {1, 2, . . . , k}, such that any two adjacent vertices and incident edges receive different colors. A total r-dynamic coloring is a proper k-coloring c of G, such that ∀v ∈ V (G), |c(N(v))| ≥ min{r, d(v) + |N(v)|} and ∀e ∈ E(G), |c(N(e))| ≥ min{r, d(v) + d(u)}. The total r-dynamic chromatic number, written as χ ”r(G), is the minimum k such that G has an r-dynamic total k-coloring. A total r-dynamic coloring is a natural extension of r-dynamic coloring in which we consider more condition of the concept, namely not only assign a color on the vertices as well as on the edges. Consequently, this study will be harder. In this paper, we will initiate to analyze a total r-dynamic of an edge comb product of two graphs, denoted by H D K, where H is path graph and K is any special graph. The result shows that the total r-dynamic chromatic number of Pn D K.