Graphs of Large Linear Size Are Antimagic

Given a graph and a colouring , 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 , the graph G is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than K₂ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least for some constant c; we improve on this result, proving the conjecture for graphs with average degree at least some constant d₀.