Graphs of large linear size are antimagic

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 $f:E\mapsto {1,\dots,|E|}$,thegraph$G$ 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 $c\log \mathrm{\mid }V\mathrm{\mid }$ for some constant $c$; we improve on this result, proving the conjecture for graphs with average degree at least some constant $d_0$.