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

Kusbudiono Kusbudiono (CGANT University of Jember Mathematics Depart. University of Jember)
Desi Febriani Putri (Mathematics Depart. University of Jember)
Dafik Dafik (CGANT University of Jember Mathematics Edu. Depart. University of Jember)
Arika Indah Kristiana (CGANT University of Jember Mathematics Edu. Depart. University of Jember)

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

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.