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Antimagic Labeling of Central Digraphs Constructed from Loops

Raja Rawat

arXiv:2610.35419v1cs.DMmath.CO

TL;DR

The paper examines antimagic labeling for central digraphs arising from loops, connecting graph-labeling questions with the algebraic framework of central digraphs. It gives a lexicographic consecutive arc-labeling and proves it is antimagic for every loop of order k ≥ 3, with the property depending only on k.

  • Problem

    The paper studies antimagic labeling in central digraphs constructed from loops, bringing graph-labeling questions together with the algebraic framework of central digraphs.

    Source: §1 · pp. 1–2

  • Method

    It orders the loopless central digraph’s arcs lexicographically by tail and head, labels them consecutively, and proves the resulting vertex weights are pairwise distinct.

    Source: §1 · p. 2, §3 · pp. 7–8

  • Results

    For every loop of order k ≥ 3, the consecutive labeling is antimagic on the loopless central digraph, and the property depends only on k.

    Source: §1 · p. 2, §3 · p. 7, §4 · p. 9

  • Takeaways & Limitations

    Non-isomorphic loops of the same order yield loopless central digraphs that are antimagic under the same explicit labeling.

    Source: §4 · p. 9

    The stated result covers loops of order k ≥ 3; the setup specifies identity 0.

    Source: §2 · p. 2

Abstract

from arXiv · show

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.

1 Introduction

The paper brings antimagic labeling to central digraphs constructed from loops, drawing together graph-labeling and algebraic frameworks. It proves that an explicit consecutive labeling makes the loopless digraph antimagic for every loop of order k ≥ 3.

1 Introduction

  • Antimagic labeling assigns labels 1 through m bijectively so that the vertex sums are pairwise distinct.The paper describes earlier results for graphs with minimum degree at least c log n, regular graphs of odd degree, and graphs with sufficiently large average degree.Source: §1 · p. 1
  • A central groupoid satisfies (x∗y)∗(y∗z) = y, and its associated digraph has adjacency matrix A satisfying A^2 = J_n.Finite central groupoids have order k^2, and every vertex in the associated digraph has in-degree and out-degree k.Source: §1 · p. 2
  • The loopless central digraph is obtained by deleting the k diagonal self-loops; non-diagonal vertices retain in-degree and out-degree k, while diagonal vertices have degree k − 1.The paper uses the associated left quasigroup with left identity and right zero to construct the digraph from a loop.Source: §1 · p. 2
  • For every loop of order k ≥ 3, lexicographically labeling the loopless central digraph’s arcs consecutively gives an antimagic labeling.The digraph has k^2 vertices and k^3 − k arcs.Source: §1 · p. 2

2 Setup and Notation

The setup fixes a loop of order k ≥ 3 and its associated left quasigroup, then labels arcs in lexicographic order. The proof uses the digraph’s degree structure and vertex sums to derive the oriented weights.

2 Setup and Notation

  • The setup takes a loop of order k ≥ 3 with identity 0 and its associated left quasigroup with left identity and right zero.The digraph is loopless after its k diagonal self-loops are removed.Source: §2 · pp. 2–3
  • Vertices are encoded as u = l1k + l2, and arcs are ordered first by tail and then by head, assigning consecutive labels 1 through m.Because this is a total order on the arc set, the labeling is bijective; arcs leaving a given tail receive consecutive labels.Source: §2 · pp. 3–4
  • The loopless digraph has k^3 − k arcs after removing k self-loops from k^3 arcs; non-diagonal vertices retain in- and out-degree k, while diagonal vertices have in- and out-degree k − 1.Source: §2 · pp. 3–4
  • The labeling’s tail offset counts earlier outgoing arcs, accounting for each earlier diagonal vertex’s one-arc degree reduction.The diagonal count before u is used to express the offset.Source: §2 · pp. 3–4
  • For a diagonal vertex (a, a), the actual oriented weight differs from its corresponding non-diagonal formula by k − 1 − a.The section derives oriented weights by subtracting the in-sum from the out-sum.Source: §2 · pp. 6–7

3 Proof of Main Theorem 1.1

The proof establishes distinct oriented vertex weights by analyzing consecutive vertices in the row-by-row ordering. It first proves that out-sums are pairwise distinct, then compares weight changes across transition cases.

3 Proof of Main Theorem 1.1

  • Pairwise distinct out-sums follow from checking consecutive differences across the three degree-transition cases.When both consecutive vertices are non-diagonal, the out-sum difference is k^2 > 0.Source: §3 · pp. 7–8
  • Each vertex’s in-sum uses exactly d_u labels, one from each in-arc’s label block.The in-arcs are determined by the edge rules for vertices with second coordinate zero or nonzero.Source: §3 · p. 8
  • The proof checks four types of consecutive-vertex transitions to establish distinct oriented weights.The cases are determined by diagonal status and whether the sequence crosses a row boundary.Source: §3 · p. 8
  • For non-diagonal-to-diagonal and diagonal-to-nondiagonal transitions, the oriented-weight change is strictly negative in the stated ranges.The respective bounds are at most −5 and at most −2.Source: §3 · p. 9

4 Consequences

The antimagic property is uniform across loops of a fixed order: it depends on k, not the loop’s isomorphism class. The proof applies the same labeling to every loop of that order.

4 Consequences

  • The antimagic property depends only on k, so non-isomorphic loops of the same order have antimagic associated digraphs under the same explicit labeling.The out-sums are the same for each loop of order k, and the distinctness argument applies uniformly.Source: §4 · p. 9
  • The uniformity follows because the proof’s out-sums agree across loops of order k, while its distinctness and weight arguments apply to all of them.Source: §4 · p. 9

Worked Example: Antimagic Labeling for k = 3

A worked example uses k = 3 to illustrate the theorem’s steps with concrete numbers before the general argument.

Worked Example: Antimagic Labeling for k = 3

S and its arc set

For k = 3, vertices are encoded as u = 3a + b, and the loopless central digraph deletes self-loops at diagonal vertices. The verification computes out-sums, in-sums, and weights, showing strict weight decrease and loop-independent weights for order 3.

S and its arc set

  • For k = 3, the nine vertices are encoded by u = 3a + b, and the loopless digraph deletes diagonal self-loops from the lexicographically ordered arc list.The table marks diagonal vertices and labels the 24 arcs.Source: S and its arc set · p. 10, Table 1 · p. 10

Step 1: Computing the out-sums S−(u)

Step 3: Computing the in-sums S+(u)

Verification of the in-sum formulas

Step 4: Computing the weights w(u) and verifying strict decrease

Observations

  • The weights are strictly decreasing and antisymmetric about the midpoint; for k = 3, all nine are distinct and the same sequence occurs for every loop of order 3.The out-sums need not be monotone, while the weights are; the unique zero weight is at the middle vertex.Source: Observations · p. 13
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