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Vertex-Coloring 2-Edge-Weighting of Graphs

Abstract

A kk-{\it edge-weighting} ww of a graph GG is an assignment of an integer weight, w(e)∈{1,…,k}w(e)\in \{1,\dots, k\}, to each edge ee. An edge weighting naturally induces a vertex coloring cc by defining c(u)=βˆ‘u∼ew(e)c(u)=\sum_{u\sim e} w(e) for every u∈V(G)u \in V(G). A kk-edge-weighting of a graph GG is \emph{vertex-coloring} if the induced coloring cc is proper, i.e., c(u)β‰ c(v)c(u) \neq c(v) for any edge uv∈E(G)uv \in E(G). Given a graph GG and a vertex coloring c0c_0, does there exist an edge-weighting such that the induced vertex coloring is c0c_0? We investigate this problem by considering edge-weightings defined on an abelian group. It was proved that every 3-colorable graph admits a vertex-coloring 33-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In particular, we show that 3-connected bipartite graphs admit vertex-coloring 2-edge-weighting

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