The antimagic orientation problems for graphs obtained by some graph operations

Abstract

A simple graph G is said to admit an antimagic orientation if there exist an orientation on the edges of G and a bijection from E(G) to {1, 2, . . . , |E(G)|} such that the vertex sums of vertices are pairwise distinct, where the vertex sum of a vertex is defined to be the sum of the labels of the in-edges minus the that of the outedges incident to the vertex. It was conjectured by Hefetz, Mütze, and Schwartz [5] in 2010 that every connected simple graph admits an antimagic orientation. In this note, we prove that the Mycielski construction of any simple graph with at most one isolated vertex admits an antimagic orientations, regardless of whether the original graph admits an antimagic orientation.