Local antimagic chromatic number of partite graphs

TL;DR

This paper investigates the local antimagic chromatic number of specific tripartite and bipartite graphs, providing exact values for these classes based on graph parameters.

Contribution

It determines the local antimagic chromatic number for complete tripartite graphs and multiple copies of bipartite graphs with certain parity conditions.

Findings

Exact local antimagic chromatic number for $K_{1,m,n}$

Determination of chromatic number for multiple copies of $K_{m,n}$

Results depend on parity conditions of $m$ and $n$

Abstract

Let $G$ be a connected graph with $|V| = n$ and $|E| = m$. A bijection $f:E\rightarrow \{1,2,...,m\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, $w(u) \neq w(v)$, where $w(u) = \sum_{e \in E(u)}f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color $w(v)$. The local antimagic chromatic number is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. Let $m,n > 1$. In this paper, the local antimagic chromatic number of a complete tripartite graph $K_{1,m,n}$, and $r$ copies of a complete bipartite graph $K_{m,n}$ where $m \not \equiv n \bmod 2$ are determined.