Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach

2024/03/25 by Lau, Gee-Choon, Shiu, Wai Chee, Premalatha, K. +1
#05C69 #05C78 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.16484

Abstract

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if there is a bijection f:E →\1,… ,|E|\ such that for any pair of adjacent vertices x and y, f⁺(x)\not= 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 χ_la(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give three ways to construct a (3m+2)× (2k+1) matrix that meets certain properties for m=1,3 and k≥ 1. Consequently, we obtained many (disconnected) graphs of size (3m+2)(2k+1) with local antimagic chromatic number 3.

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