Vertex-coloring graphs with 4-edge-weightings

TL;DR

This paper extends the known results on vertex-coloring edge-weightings by proving that graphs without isolated edges can be vertex-colored with weights from {1,2,3,4}, supporting the conjecture for a broader set.

Contribution

It proves that the vertex-coloring edge-weighting conjecture holds when using the weight set {1,2,3,4}, expanding previous results that used smaller sets.

Findings

The conjecture is true for weight set {1,2,3,4}.

Graphs without isolated edges can be properly vertex-colored with these weights.

Supports the broader conjecture for larger weight sets.

Abstract

An edge-weighting of a graph is called vertex-coloring if the weighted degrees yield a proper vertex coloring of the graph. It is conjectured that for every graph without isolated edge, a vertex-coloring edge-weighting with the set {1,2,3} exists. In this note, we show that the statement is true for the weight set {1,2,3,4}.