The local antimagic total vertex coloring of graphs with homogeneous pendant vertex

Abstract

A graph consists vertices and edges. As usual, the vertex set denoted by V and edge set by E. Graph G(V, E) be connected and undirected graphs. A graph G called homogeneous pendant vertex if the graph G have pendant edge. The concept of local antimagic coloring of a graph was introduced by Arumugam et.al (2017). Thus, we initiate to developed the concept of local antimagic namely local antimagic total vertex coloring. Local vertex antimagic total coloring is defined f : V(G) ⋃ E(G) → {1, 2, 3…, |V(G)| + |E(G)|} if for any two adjacent vertices v1 and v2, w(v1) ≠ w(v2), where for v ∈ G, w(v) = ∑e∈E(v) f(e) + f(v), where E(v) and V(v) are respectively the set of edges incident to v and the set of vertices adjacent to v. Thus, the local antimagic total vertex labeling induces a proper vertex coloring of G if each vertex v is assigned the color wt(v).The minimum colors needed to coloring the vertices of a graph G called local antimagic total vertex chromatic number of graph G and denoted by χlatv(G). In this paper we study the local antimagic total vertex coloring of graph G ⊙ mK1 and determined exact value of graphs as follows Sn ⊙ mK1, Pn ⊙ mK1, Cn ⊙ mK1 and Fn ⊙ mK1.