Vertex-Colouring Edge-Weightings

A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted c _v , is ${\sum\nolimits_{e \mathrel\backepsilon v} {w{\left( e \right)}} }$. We show that the edges of every graph that does not contain a component isomorphic to K₂ can be weighted from the set {1, . . . ,30} such that in the resulting vertex-colouring of G, for every edge (u,v) of G, c _u ≠c _v .