Antimagicness of graphs with a dominating clique

TL;DR

This paper investigates antimagic labelings in graphs with a dominating clique, introducing a new C-antimagic concept and proving that most such graphs are 3-antimagic, expanding understanding of antimagic graph properties.

Contribution

It extends antimagicness results to graphs with a dominating clique and introduces C-antimagic labelings, showing most such graphs are 3-antimagic.

Findings

Most graphs with a dominating clique are 3-antimagic.

Introduces C-antimagic labeling as a generalization.

Extends previous antimagicness results to new graph classes.

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.