Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

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.

Chromatic number of super vertex local antimagic total labelings of graphs

“…Slamin et al [8] proved the following. If T is a tree on n ≥ 2 vertices with k leaves, then χ slat (T ) ≤ n − k + 1.…”

“…We will show that stars are the only trees with χ slat (T ) = 2. In our proof, we provide a labeling different from the one in [8].…”

“…Furthermore, Slamin et al [8] introduced a new variant of the labeling. A super vertex local antimagic total labeling is a bijective map f : From the definition, we can perceive the super vertex local antimagic labeling as a vertex coloring of a graph with some additional conditions.…”

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

“…To tackle these problems, different coloring schemes have been proposed: the scheme based on distances in [44], the scheme based on templates in [45], the scheme based on adjacencies in [46], the scheme based on heuristics in [47] and the scheme based on pseudo-randomness (with constrains, Grundy and color-dominating) in [48]. The properties of the colorings have been studied in [49] and the counting of distinguishing (symmetry breaking) colorings with k colors in [50].…”

An Efficient Shortest Path Algorithm: Multi-Destinations in an Indoor Environment

“…Nowadays, many researchers are working on graphical solutions and pathfinding to resolve crucial application issues [21][22][23][24][25][26][27]. Dijkstra's algorithm is used for finding the solution to the shortest path problem in multi-destination nodes [28].…”