L-factors and adjacent vertex-distinguishing edge-weighting

TL;DR

This paper explores the relationship between edge-weighting problems and graph factors, proving that all 4-colorable graphs can be vertex-colored with 4-edge weights, thus advancing understanding of graph coloring and weighting.

Contribution

It establishes new results on the existence of specific graph factors and applies these to prove that every 4-colorable graph admits a 4-edge-weighting for vertex-coloring.

Findings

Every 4-colorable graph admits a vertex-coloring 4-edge-weighting.

Generalizes earlier results on graph factors with prescribed degrees.

Links edge-weighting problems to special factors of graphs.

Abstract

An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting problem has a close relationship with special factors of graphs. In this paper, we obtain several results on the existence of factors with the pre-specified degrees, which generalizes earlier results in [2, 3]. Using these results, we investigate edge-weighting problem. In particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weighting.