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.

A note on local edge antimagic chromatic number of graphs

“…The study is followed by Rajkumar and Nalliah [15] who investigated the local edge antimagic chromatic number of friendship graphs F n , wheels W n , fan graphs f n , helm graph H n , and flower graphs F l n . Many variations on local antimagic labeling may also be seen in [5,8,9]. To see many other kinds of labeling please consult to [6].…”