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

### Abstract

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.
Published

2017-08-08

How to Cite

WARDANI, Dwi Agustin Retno et al.
On The Total r-dynamic Coloring of Edge Comb Product graph G D H.

**UNEJ e-Proceeding**, [S.l.], p. 325-327, aug. 2017. Available at: <https://jurnal.unej.ac.id/index.php/prosiding/article/view/4256>. Date accessed: 13 aug. 2024.
Section

General