Local antimagic vertex total coloring on fan graph and graph resulting from comb product operation

Abstract

Let G = (V,E) be a connected graph with |V|=n and |E|=m. A bijection f: V(G) U E(G) → {1,2,3, …,n + m} is called local antimagic vertex total coloring if for any two adjacent vertices u and v, wt(u)≠ wt(v), where wt(u) = ∑e∈E(u) f(e) + f(u), and E (u) is a set of edges incident to u. Thus any local antimagic vertex total labeling induces a proper vertex coloring of G where the vertex v is assigned the color wt (v). The local antimagic vertex total chromatic number χ1να t(G) is the minimum number of colors taken over all colorings induced by local antimagic vertex total. In this paper we investigate local antimagic vertex total coloring on fan graph (Fn) and graph resulting from comb product operation of Fn and F3 which denoted by Fnt> F3. We get two theorems related to the local antimagic vertex total chromatic number. First, χ1να t(Fn) = 3 where n> 3. Second, 3 < χ1ναt(Fn > F3) < 5 where n > 3.