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