On Local Antimagic Chromatic Number of Spider Graphs

TL;DR

This paper investigates the local antimagic chromatic number of spider graphs, establishing bounds, conditions for attainability, and exact values for specific cases, contributing to graph labeling theory.

Contribution

It provides bounds and conditions for the local antimagic chromatic number of spider graphs, including exact values for 3-leg spiders and a conjecture for general cases.

Findings

For d-leg spiders, the chromatic number is between d+1 and d+2.

3-leg spiders have chromatic number 4 unless all legs are odd.

Conjecture: most d-leg spiders with certain size conditions have chromatic number d+1.

Abstract

An edge labeling of a connected graph $G = (V,E)$ is said to be local antimagic if it is a bijection $f : E \to \{1, . . . , |E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \ne f^+(y)$, 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 $G$. In this paper, we first show that a $d$-leg spider graph has $d+1\le \chi_{la}\le d+2$. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has $\chi_{la} = 4$ if not all legs are of odd length. We conjecture that almost all $d$-leg spiders of size $q$ that satisfies $d(d+1) \le 2(2q-1)$ with each leg length at least 2 has $\chi_{la} = d+1$.