# On Total r-Dynamic Coloring of Several Classes of Graphs and Their Related Operations

### Abstract

All graphs in this paper are simple, connected and undirected. Let r, k be natural numbers. By a proper k-coloring of a graph G, we mean a map c : V (G) → S, where |S| = k, such that any two adjacent vertices 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 ∀uv ∈ E(G), |c(N(uv))| ≥ min[r, d(u) + d(v)]. The total r-dynamic chromatic number, written as χ ′′r (G), is the minimum k such that G has an r-dynamic k-coloring. Finding the total r-dynamic chromatic number is considered to be a NP-Hard problems for any graph. Thus, in this paper, we initiate to study χ′′ r (G) of several classes of graphs and and their related operations.
Published

2017-08-08

How to Cite

KUSBUDIONO, Kusbudiono et al.
On Total r-Dynamic Coloring of Several Classes of Graphs and Their Related Operations.

**UNEJ e-Proceeding**, [S.l.], p. 331-336, aug. 2017. Available at: <https://jurnal.unej.ac.id/index.php/prosiding/article/view/4258>. Date accessed: 08 aug. 2024.
Section

General