Weighted bipartite edge colouring
Let G=(S,T;E) be a bipartite graph, with weights [math]w:E \to [0,1][/math]. A proper weighted edge colouring is a colouring of the edges such that at each vertex, the sum of weights of edges of the same colour is at most 1. A lower bound for the number of colours is the minimum number of unit bins needed to pack the weights incident to any vertex, denoted by b. Is there always a proper weighted edge colouring using [math]2b-1[/math] colours?