Vertex-colouring edge-weightings (Q2466341)

A weighting \(w\) of the edges of a graph \(G\) induces a colouring of the vertices of \(G\), where the colour \(c_v\) of a vertex \(v\) is defined by \(\sum_{v\in e}w(e)\). The authors prove that the edges of every graph \(G\) without a component isomorphic to the complete graph \(K_2\) can be weighted from the set \(\{1,2,\dots,30\}\) such that \(c_u\neq c_v\) in the resulting vertex-colouring of \(G\) for every edge \(uv\) of \(G\).