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Antimagicness of graphs with a dominating clique

Grégoire Beaudoire, Cédric Bentz, Christophe Picouleau

TL;DR

This work advances antimagic labeling by focusing on graphs with a dominating clique and introducing the C-antimagic relaxation. It proves antimagicness under a degree-bound condition and establishes 3-antimagicness under a quantified edge-count bound through a staged edge-partition labeling that carefully controls vertex-sums. The methods rely on partitioning edges into structured sets and using maximal-path labelling plus post-processing swaps to resolve potential collisions in vertex sums. The authors also propose a broad conjecture that every connected graph (excluding K2) is C-antimagic for some constant C, outlining open problems and directions for extending these results to other dominating-clique configurations and degree regimes.

Abstract

A graph $G = (V, E)$ is called antimagic if there exists a bijective labelling $f : E \rightarrow \{1, 2, \ldots, |E|\}$ such that the vertex-sums of labels over edges incident to a given vertex are all distinct. In this paper, we extend the antimagicness results over graphs with a dominating clique. We also introduce an alternative to the usual definition of antimagic graphs, called C-antimagic, allowing for the labelling to be injective in $\{1, 2, . . . , |E| + C\}$ instead of bijective, and show that almost all graphs with a dominating clique are 3-antimagic.

Antimagicness of graphs with a dominating clique

TL;DR

This work advances antimagic labeling by focusing on graphs with a dominating clique and introducing the C-antimagic relaxation. It proves antimagicness under a degree-bound condition and establishes 3-antimagicness under a quantified edge-count bound through a staged edge-partition labeling that carefully controls vertex-sums. The methods rely on partitioning edges into structured sets and using maximal-path labelling plus post-processing swaps to resolve potential collisions in vertex sums. The authors also propose a broad conjecture that every connected graph (excluding K2) is C-antimagic for some constant C, outlining open problems and directions for extending these results to other dominating-clique configurations and degree regimes.

Abstract

A graph is called antimagic if there exists a bijective labelling such that the vertex-sums of labels over edges incident to a given vertex are all distinct. In this paper, we extend the antimagicness results over graphs with a dominating clique. We also introduce an alternative to the usual definition of antimagic graphs, called C-antimagic, allowing for the labelling to be injective in instead of bijective, and show that almost all graphs with a dominating clique are 3-antimagic.

Paper Structure

This paper contains 4 sections, 6 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction and definitions
  2. Proof of Theorem \ref{['dominanted']}
  3. Proof of Theorem \ref{['injectioned']}
  4. Conclusion and future work

Key Result

Theorem 1

Let $G = (V,E)$ be a graph with at least $3$ vertices. If $G$ has a clique $B$ such that, for every vertex $v \in V$, either $N(v) \subset B$ or $B \subset N(v)$, then $G$ is antimagic.

Figures (1)

  • Figure 1: Illustration of the structure of graphs described in Theorem \ref{['barrused']}: $A$ is an independent set made of vertices $u$ such that $N(u) \subset B$, and $C$ is the set of vertices $v$ such that $B \subset N(v)$ (with no particular structure).

Theorems & Definitions (15)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1
  • Theorem 5
  • proof
  • Theorem 5
  • proof
  • ...and 5 more