On the super edge local antimagic total labeling of related ladder graph

Abstract

A graph G is an ordered pair of sets G = (V, E), where V is a set of vertices and E is a set of edges. The concept of labeling graphs has recently gained a lot of papularity in the area of graph theory. Antimagic labeling is the interest topic to studied. The study of antimagic labeling motivated by Hartsfield and Ringel. Arumugam et.al developed the concept of antimagic labeling that is local antimagic coloring. Thus, we study the concept of local antimagic namely edge local antimagic total labeling. By an edge local antimagic total labeling, a bijection f : V(G) ∪ E(G) → {1, 2, 3, …, |V(G)| + |E(G)|} satisfying that for any two adjacent edges e 1 and e 2, wt (e 1) ≠ wt (e 2), where for e = ab ∈ G, wt (e) = f(a) + f(b) + f(ab). Thus, any edge local antimagic total labeling induces a proper edge coloring of G if each edge e is assigned the color wt (e). It is considered to be a super edge local antimagic total coloring, if the smallest labels appear in the vertices. The chromatic number of super edge local antimagic total, denoted by γleat (G), is the minimum number of colors taken over all colorings induced by super edge local antimagic total labelings of G. In this paper we study edge local antimagic total labeling and determined the chromatic number of graphs as follow grid graph, prism graph and mobious ladder graph.