On The Total r-dynamic Coloring of Edge Comb Product graph G D H - Dwi Agustin Retno Wardani; Dafik Dafik; Antonius C. Prihandoko; Arika I. Kristiana (UPT Penerbitan Universitas Jember, 2016)
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))| ≥ minr, d(v) + |N(v)| and ∀e ∈ E(G), |c(N(e))| ≥ minr, 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.