Chromatic number of super vertex local antimagic total labelings of graphs

Abstract

Let G (V, E) be a simple graph and f be a bijection f: V ∪ E → 1, 2, …, |V|+ |E| where f (|V|) = 1, 2, …, |V|. For a vertex x ∈ V, define its weight w (x) as the sum of labels of all edges incident with x and the vertex label itself. Then f is called a super vertex local antimagic total (SLAT) labeling if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χslat (G) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. We classify all trees T that have χslat (T) = 2, present a class of trees that have χslat (T) = 3, and show that for any positive integer n ≥ 2 there is a tree T with χslat (T) = n.