On local antimagic chromatic numbers of the join of two special families of graphs

TL;DR

This paper determines the local antimagic chromatic number for joins of specific graph families using matrix methods, revealing infinitely many tripartite graphs with chromatic number 3.

Contribution

It introduces a matrix-based approach to exactly compute local antimagic chromatic numbers for joins of null and 1-regular graphs, expanding understanding of graph colorings.

Findings

Complete determination of local antimagic chromatic number for joins of null and 1-regular graphs.

Identification of infinitely many tripartite graphs with chromatic number 3.

Method applicable to possibly disconnected or regular graphs.

Abstract

It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we use matrices of size $(2m+1) \times (2k+1)$ to completely determine the local antimagic chromatic number of the join of null graphs, $O_m, m\ge 1,$ and 1-regular graphs of odd components, $(2k+1)P_2$, $k\ge 1$. Consequently, we obtained infinitely many (possibly disconnected or regular) tripartite graphs with local antimagic chromatic number 3.