Antimagic labeling proven for certain loop-based central digraphs
Antimagic Labeling of Central Digraphs Constructed from Loops
Discrete Mathematics
Summary
The paper studies a special way to assign numbers to the connections (arcs) between points (vertices) in a certain type of directed graph made from mathematical loops. They show explicitly how to label these arcs so that each point gets a unique sum of incoming and outgoing arc labels, which is called an antimagic labeling. This property holds for all such graphs created from loops with at least three elements, regardless of the specific structure of the loop. Essentially, they provide a new proof that all these graphs have this unique numbering property.
What this means in practice
- •For network designers: Design directed networks with uniquely identifiable nodes using antimagic arc labeling to improve routing and fault detection.
- •For coding theory engineers: Develop combinatorial structures for error detection and correction codes based on antimagic properties of loop-derived digraphs.
A theory result. No direct application yet.
Authors
Raja Rawat
Abstract
Let $(T, \circ)$ be a loop of order $k \geq 3$ and $D_S^*$ the loopless central digraph obtained from the associated central digraph $D_S$ by removing the $k$ self-loops at the diagonal vertices. We construct an explicit arc-labeling $f$ of $D_S^*$ ordering all arcs lexicographically by $(\text{tail}, \text{head})$ and assigning labels $1, \ldots, m$ in that order and prove that $f$ is an antimagic labeling. Consequently, $D_S$ is antimagic for every loop of order $k \geq 3$, a property that depends only on $k$ and is invariant across all non-isomorphic loops of the same order.