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On a \(1,2\) conjecture. (English) Zbl 1250.05093
Summary: Let us assign positive integers to the edges and vertices of a simple graph \(G\). As a result we obtain a vertex-colouring of \(G\) with integers, where a vertex colour is simply a sum of the weight assigned to the vertex itself and the weights of its incident edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary \(G\)? We give a positive answer when \(G\) is a 3-colourable, complete or 4-regular graph. We also show that it is enough to use weights from 1 to 11, as well as from 1 to \(\lfloor \chi (G) / 2\rfloor +1\), for an arbitrary graph \(G\).

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C15 Coloring of graphs and hypergraphs
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