Majority Edge-Colorings of Graphs

TL;DR

This paper introduces the concept of majority edge-colorings in graphs, establishing bounds on the minimum degree for such colorings with 3 or 4 colors, and discusses related variations and open problems.

Contribution

The paper defines majority edge-colorings and proves that graphs with minimum degree at least 2 or 4 have such colorings with 4 or 3 colors respectively.

Findings

Graphs with minimum degree ≥ 2 have a majority 4-edge-coloring.

Graphs with minimum degree ≥ 4 have a majority 3-edge-coloring.

Discussion of variations and open problems in majority edge-colorings.

Abstract

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.