Title

Affirmative Solutions on Local Antimagic Chromatic Number

Publication Date

9-1-2020

Document Type

Article

Department

Mathematics and Statistics

Publication Title

Graphs and Combinatorics

Volume

36

Issue

5

DOI

10.1007/s00373-020-02197-2

First Page

1337

Last Page

1354

Abstract

An edge labeling of a connected graph G= (V, E) is said to be local antimagic if it is a bijection f: E→ { 1 , … , | E| } such that for any pair of adjacent vertices x and y, f+(x) ≠ f+(y) , where the induced vertex label f+(x) = ∑ f(e) , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G) , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of χla(G∨ O2) that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of χla(G∨ On) and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.

Funding Number

14JCYBJC43100

Funding Sponsor

Universiti Teknologi MARA

Keywords

Local antimagic chromatic number, Local antimagic labeling

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