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

TL;DR

This paper introduces a matrix-based method to construct specific graphs with fixed even size and local antimagic 3-colorability, expanding the understanding of graph labelings and chromatic properties.

Contribution

It presents a novel matrix approach to construct bipartite and tripartite graphs of fixed even size with local antimagic chromatic number 3, providing new families of such graphs.

Findings

Constructed matrices with integers in [1, 10k] satisfying certain properties.

Generated many families of bipartite and tripartite graphs of size 10k.

Achieved graphs with local antimagic chromatic number 3.

Abstract

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. Suppose $\chi_{la}(G)=\chi_{la}(H)$ 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 $[1,10k]$, $k\ge 1$, that meet certain properties. Consequently, we obtained many families of (disconnected) bipartite (and tripartite) graphs of size $10k$ with local antimagic chromatic number 3.