On Local Antimagic Vertex Coloring for Complete Full t-ary Trees

Abstract

Let G = (V, E) be a finite simple undirected graph without K2 components. A bijection f : E → {1, 2, ⋯, |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) ≠ w(v), where the vertex sum w(u) = ∑e∈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 χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. It was conjectured [6] that for every tree T the local antimagic chromatic number l + 1 ≤ χla(T) ≤ 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 ≥ 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.