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 χ_slat(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 χ_slat(T)=2, present a class of trees that have χ_slat(T)=3, and show that for any positive integer n ≥ 2 there is a tree T with χ_slat(T)=n.