A note on (a, d)-vertex antimagic total labeling of paths and cycles

Abstract

A labeling of a graph G(V, E) is an (a, d)-vertex antimagic total labeling (VATL) if it assigns the vertices and edges the consecutive integers from 1 to vertical bar V vertical bar + vertical bar E vertical bar with the property that the sums for each vertex of its label and the labels on its incident edges form an arithmetic progression with initial term a and common difference d. We give some positive and negative results on the existence of VATLs of paths and cycles.