Antimagic labelling of graphs with maximum degree $\Delta(G) = n - 4$

TL;DR

This paper extends the class of graphs known to be antimagic by proving that graphs with maximum degree n-4 are antimagic if they have at least 7n edges, broadening previous results.

Contribution

It proves that graphs with maximum degree n-4 are antimagic under the condition that the number of edges is at least 7 times the number of vertices.

Findings

Graphs with Δ(G) = n - 4 are antimagic if |E| ≥ 7n.

Extends previous results from Δ(G) ≥ n - 3 to Δ(G) = n - 4.

Provides new conditions under which graphs are antimagic.

Abstract

An antimagic labelling of a graph $G = (V,E)$ is a bijection from $E$ to $\{1,2, \ldots, |E|\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is said to be antimagic if it has an antimagic labelling. It has been proven that graphs $G$ with $\Delta(G) \geq n - 3$ are antimagic, where $\Delta(G)$ is the maximum degree of a vertex in $G$ and $n = |V|$. In this article, we extend this result to graphs with $\Delta(G) = n - 4$, provided that $|E| \geq 7n$.