Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach

2024/04/28 by Gee-Choon Lau, Wai Chee Shiu, Lau, Gee-Choon +5
Computer Science · #05C69 #05C78 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2404.18049

openalex publication_date 2024/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)\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. Suppose χ_la(G)=χ_la(H) and G_H is obtained from G and H by merging some vertices of G with some vertices of H bijectively. In this paper, we give ways to construct matrices with integers in [1,10k], k≥ 1, that meet certain properties. Consequently, we obtained many families of (disconnected) bipartite (and tripartite) graphs of size 10k with local antimagic chromatic number 3.

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