Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

Slamin Slamin,Kristiana Wijaya,Nelly Oktavia Adiwijaya,Dafik Dafik,Muhammad Ali Hasan

doi:10.3390/sym12111843

Abstract

Let G=(V,E) be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from V∪E to {1,2,…,|V|+|E|} such that if for each uv∈E(G) then w(u)≠w(v), where w(u)=∑uv∈E(G)f(uv)+f(u). If the range set f satisfies f(V)={1,2,…,|V|}, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color w(v) assigning the vertex v. The local super antimagic total chromatic number of graph G, χlsat(G) is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

Highlights

Vertex coloring is an assignment of colors to every vertex of graph G such that any two adjacent vertices receive different colors and the number of colors used for such coloring is made as minimal as possible
The k-coloring of graph G is a map of c : V → {1, 2, . . . , k} where V is the set of vertices of G and c(v) is a color of vertex v such that c(u) 6= c(v) whenever vertices u and v are adjacent
The vertex coloring of graphs is approached by using the local antimagic total labeling of a graph

Summary

Introduction

Vertex coloring is an assignment of colors to every vertex of graph G such that any two adjacent vertices receive different colors and the number of colors used for such coloring is made as minimal as possible. By using Theorem 1 and the fact that χlsat ( G ) ≥ 2 for any connected graph G of order of at least 2, we obtain the local super antimagic total chromatic number of the star as follows. We present the local super antimagic total chromatic number of a cycle Cn on n ≥ 3 vertices.

Results

Conclusion