The antimagic orientation problems for graphs obtained by some graph operations

TL;DR

This paper investigates antimagic orientations in graphs and proves that certain graph operations, like the Mycielski construction and corona product, produce graphs that satisfy the antimagic orientation conjecture.

Contribution

It demonstrates that applying specific graph operations results in graphs that admit antimagic orientations, advancing understanding of the conjecture for complex graph classes.

Findings

Mycielski construction preserves antimagic orientation property

Corona product graphs can admit antimagic orientations under certain conditions

Supports the conjecture for broader classes of graphs

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 $\{1,2,\ldots,|E(G)|\}$ 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 that of the out-edges incident to the vertex. It was conjectured by Hefetz, M\"{u}tze, and Schwartz~\cite{HMS10} in 2010 that every connected simple graph admits an antimagic orientation. In this paper, we prove that the Mycielski construction and the corona product for graphs with some conditions yield graphs satisfying the above conjecture.