Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

Abstract. A weighting of the edges of a graph is called vertexcoloring if the weighted degrees of the vertices yield a proper coloring of the graph. In this paper we show that such a weighting is possible from the weight set {1, 2, 3, 4, 5} for all graphs not containing components with exactly 2 vertices. All graphs in this note are finite and simple. For notation not defined here we refer the reader to [3]. For some k ∈ N, let ω : E(G) → {1, 2, . . . , k} be an integer weighting of the edges of a graph G. This weighting is called vertex-coloring if the weighted degrees ω(v) = u∈N(v) ω(uv) of the vertices yield a proper vertex-coloring of the graph. It is easy to see that for every graph which does not have a component isomorphic to K2 , there exists such a weighting for some k. In 2002, Karo´nski, Luczak and Thomason (see [4]) conjectured that such a weighting with k = 3 is possible for all such graphs (k = 2 is not sufficient as seen for instance in complete graphs and cy...

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Maciej Kalkowski, Michal Karonski, Florian Pfender
• Keywords: JCT 2010, Weighted Degrees, Weighting, first Constant Bound