The 1-2-3 Conjecture

1-2-3 conjecture: the edges of every graph without isolated edges can be weighted with \(\{1, 2, 3\}\) so that the weighted degrees (sums of incident weights) form a proper vertex colouring — adjacent vertices receive distinct sums.

Conjectured by Karoński, Łuczak and Thomason in 2004. The 1-2 weights alone fail in general (e.g. long paths need the third weight), and the conjecture asserts three weights always suffice. It is known for complete graphs, bipartite graphs, 3-colourable graphs (Kalkowski–Karoński–Pfender 2010, who showed {1,2,3} works there with a slick ordering argument) and many other classes — but the general graph remains open, with the list version (weights from individual lists) harder still.