Distance Antimagic Product Graphs

Abstract

A distance antimagic graph is a graph G admitting a bijection f:V(G)→{1,2,…,|V(G)|} such that for two distinct vertices x and y, ω(x)≠ω(y), where ω(x)=∑y∈N(x)f(y), for N(x) the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs.