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

Let $G=(V,E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f:E\to \{1,2,\cdots ,\mathrm{\mid }E\mathrm{\mid }\}$ is called a local antimagic labeling if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e., $w(u)\mathrm{\ne }w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$,and$E(u)$isthesetofedgesincidentto$u$. 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 $l+1\le {\chi }_{la}(T)\le l+2$, where $l$ is the number of leaves of $T$. In this article we verify the above conjecture for complete full $t$-ary trees, for $t\ge 2$. 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 $t$.