ON LOCAL ANTIMAGIC CHROMATIC NUMBER OF GRAPHS

Document Type : Original Manuscript

Author

School of Mathematics and Computer Science, Damghan University, P.O. Box 36716-41167, Damghan, Iran.

Abstract

A {\it local antimagic labeling} of a connected graph G with at least three vertices, is a bijection f:E(G)→{1,2,…,|E(G)|} such that for any two adjacent vertices u and v of G, the condition
ωf(u)≠ωf(v) holds; where ωf(u)=∑x∈N(u)f(xu). Assigning ωf(u) to u for each vertex u in V(G), induces naturally a proper vertex coloring of G; and |f| denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of G, denoted by χla(G), is defined as the minimum of |f|, where f ranges over all local antimagic labelings of G.
In this paper, we explicitly construct an infinite class  of connected graphs G such that χla(G) can be arbitrarily large while χla(G∨K2¯)=3, where G∨K2¯ is the join graph of G and the complement graph of K2. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex  coloring of a graph,  Graphs and Combinatorics 33} (2017), 275-285].

Keywords

Journal of Algebraic Systems
Volume 7, Issue 2
January 2020
Pages 245-256

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