Overview
This research addresses the existence of antimagic orientations in a specific class of graphs: those with a dominating clique. It presents a proof that such graphs always admit an antimagic orientation. This result contributes to the understanding of graph theory, specifically in the context of graph labeling and orientation problems.
Research Context
The concept of an antimagic orientation for a graph $G$ with $m$ edges involves orienting its edges and bijectively labeling them with integers from $1$ to $m$. The key condition for an orientation to be antimagic is that the oriented vertex sums, defined as the labels on in-arcs minus the labels on out-arcs for each vertex, must be pairwise distinct.
A central conjecture in this area, attributed to Hefetz, Mütze, and Schwartz, posits that every connected graph admits an antimagic orientation. The present work provides evidence supporting this conjecture by confirming its validity for graphs characterized by the presence of a dominating clique.
Findings
The study establishes that any graph containing a dominating clique necessarily admits an antimagic orientation. The core finding is a direct proof supporting this assertion.
- A graph $G$ with $m$ edges has an antimagic orientation if its edges can be oriented and bijectively labelled $1, ..., m$ such that the oriented vertex sums (labels on in-arcs minus labels on out-arcs) are pairwise distinct.
- The conjecture by Hefetz, Mütze and Schwartz states that every connected graph admits an antimagic orientation.
- This research proves that the conjecture holds for every graph with a dominating clique.