Dense graphs are antimagic

TL;DR

This paper proves that all sufficiently connected graphs, including certain complete partite graphs and high-degree graphs, are antimagic, confirming a longstanding conjecture for a broad class of graphs.

Contribution

It validates Ringel's conjecture for graphs with minimum degree logarithmic in the number of vertices, using probabilistic and combinatorial methods.

Findings

Connected graphs with minimum degree Ω(log n) are antimagic.

Complete partite graphs (except K_2) are antimagic.

Graphs with maximum degree at least n-2 are antimagic.

Abstract

An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called {\em antimagic} if it has an antimagic labeling. A conjecture of Ringel (see \cite{HaRi}) states that every connected graph, but $K_2$, is antimagic. Our main result validates this conjecture for graphs having minimum degree $\Omega(\log n)$. The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but $K_2$) and graphs with maximum degree at least $n-2$ are antimagic.