Local Antimagic Chromatic Number for Copies of Graphs

Andrea Semaničová-Feňovčíková,Martin Bača,Tao-Ming Wang

doi:10.3390/math9111230

Andrea Semaničová-Feňovčíková, Martin Bača

Open Access

https://doi.org/10.3390/math9111230

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Journal: Mathematics	Publication Date: May 27, 2021
Citations: 8	License type: CC BY 4.0

Affiliation: Technical University of Košice, Tunghai University

Abstract

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

Highlights

In this paper, we will consider only finite graphs without loops or multiple edges
Perhaps the most remarkable result to date is the proof for regular graphs of odd degree given by Cranston et al in [11], which was subsequently adapted to regular graphs of even degree by Bércz et al in [12] and by Chang et al in [13]
We investigate the local antimagic chromatic number for disjoint union of multiple copies of a graph G, denoted by mG, m ≥ 1, and we present some estimations of this parameter

Summary

Introduction

We will consider only finite graphs without loops or multiple edges. For graph theoretic terminology we refer to the book by Chartrand and Lesniak [1]. The concept of antimagic labeling was introduced by Hartsfield and Ringel [2] who conjectured that every simple connected graph, other than K2, is antimagic. This conjecture is still open for some special classes of graphs it was proved, see for instance [3,4,5,6,7,8]. Any local antimagic labeling induces a proper vertex coloring of G where the vertexweight wt(u) is the color of u This naturally leads to the concept of a local antimagic chromatic number introduced in [14].

Graphs with Vertices of Even Degrees

Conclusions