On vertex-coloring edge-weighting of graphs

Abstract

A k-edge-weightingw of a graph G is an assignment of an integer weight, w(e) ∈ {1, …, k}, to each edge e. An edge-weighting naturally induces a vertex coloring c by defining for every u ∈ V (G). A k-edge-weighting of a graph G is vertex-coloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). When k ≡ 2 (mod 4) and k≥6, we prove that if G is k-colorable and 2-connected, δ(G)≥k - 1, then G admits a vertex-coloring k-edge-weighting. We also obtain several suffcient conditions for graphs to be vertex-coloring k-edge-weighting.

Keywords

Vertex coloring / edge-weighting

References

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg