On local antimagic total chromatic number of certain one point union of graphs

TL;DR

This paper investigates the local antimagic total chromatic number of certain graphs, providing exact values for paths, spiders with short legs, and demonstrating existence results for unicyclic and bicyclic graphs.

Contribution

It corrects and extends previous results by determining the exact local antimagic total chromatic number for specific graph classes, including paths, spiders, and certain cyclic graphs.

Findings

Exact chromatic number for path graphs

Chromatic number for spider graphs with short legs

Existence of unicyclic and bicyclic graphs with chromatic number 3

Abstract

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w(u)\ne w(v)$, where $w(u) = f(u) + \sum_{e\in E(u)} f(e)$ and $E(u)$ is the set of incident edge(s) of $u$. The local antimagic total chromatic number, denoted $\chi_{lat}(G)$, is the minimum number of distinct weights over local antimagic total labeling of $G$. In this paper, we provide a correct proof and exact local antimagic total chromatic number of path and spider graphs given in [Local vertex antimagic total coloring of path graph and amalgamation of path, {\it CGANT J. Maths Appln.} {\bf 5(1)} 2024, DOI:10.25037/cgantjma.v5i1.109]. Further, we determined the local antimagic total chromatic number of spider graph with each leg of length at most 2. We also showed the existence of unicyclic and bicyclic graphs with local antimagic total chromatic number 3.