Super local edge antimagic total coloring of Pn ▹;h

Abstract

In this paper, we consider that all graphs are finite, simple and connected. Let G(V;E) be a graph of vertex set V and edge set E. A bijection f : V (G) → {1; 2; 3; ⋯ |V (G)|} is called a local edge antimagic labeling if for any two adjacent edges e1 and e2, w(e1) ≠ w(e2), where for e = uv 2 G, w(e) = f(u) + f(v). Thus, any local edge antimagic labeling induces a proper edge coloring of G if each edge e is assigned the color w(e). It is considered to be a super local edge antimagic total coloring, if the smallest labels appear in the vertices. The super local edge antimagic chromatic number, denoted by γleat(G), is the minimum number of colors taken over all colorings induced by super local edge antimagic total labelings of G. In this paper we initiate to study the existence of super local edge antimagic total coloring of comb product of graphs. We also analyse the lower bound of its local edge antimagic chromatic number. It is proved that γleat(Pn .G) ≤ γleat(Pn)+γleat(G). Furthermore we have determine exact value local edge antimagic coloring of Pn . Pm, Pn . Cm and Pn . Sm.