Graphs of large linear size are antimagic

TL;DR

This paper proves that all large connected graphs with sufficiently high average degree are antimagic, advancing the longstanding conjecture that all such graphs, except K2, are antimagic.

Contribution

The paper improves previous results by proving the antimagic conjecture for graphs with average degree above a certain constant, not just high minimum degree.

Findings

Proves antimagic property for graphs with average degree above a constant

Extends the class of graphs known to be antimagic beyond high minimum degree cases

Supports the conjecture that all connected graphs except K2 are antimagic

Abstract

Given a graph $G=(V,E)$ and a colouring $f:E\mapsto \mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called antimagic. If $G$ has a bijective antimagic colouring $f:E\mapsto \{1,\dots,|E|\}$, the graph $G$ is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than $K_2$ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least $c \log |V|$ for some constant $c$; we improve on this result, proving the conjecture for graphs with average degree at least some constant $d_0$.