On local antimagic vertex coloring for complete full $t$-ary trees

TL;DR

This paper investigates the local antimagic vertex coloring of complete full t-ary trees, confirming a conjecture and determining the exact chromatic number for these trees, especially for odd t.

Contribution

It verifies the conjecture on the local antimagic chromatic number for complete full t-ary trees and finds the exact value for odd t.

Findings

The local antimagic chromatic number of complete full t-ary trees is l+1 for odd t.

The paper confirms the conjecture that for trees, the chromatic number is between l+1 and l+2.

Provides explicit values for the chromatic number in the case of complete full t-ary trees.

Abstract

Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e., $w(u) \neq w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$, 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 (vertex sum) $w(v)$. The local antimagic chromatic number $\chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. It was conjectured \cite{Aru-Wang} that for every tree $T$ the local antimagic chromatic number $l+ 1 \leq \chi_{la} ( T )\leq l+2$, where $l$ is the number of leaves of $T$. In this article we verify the above conjecture for complete full $t$-ary trees, for $t \geq 2$. A complete full $t$-ary tree is a rooted tree in which all nodes have exactly $t$ children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full $t$-ary trees is $ l+1$ for odd $t$.