On degree-colorings of multigraphs

TL;DR

This paper introduces a new concept called degree-coloring for multigraphs, proposes a conjecture relating it to maximum degree and density, and proves the conjecture's equivalence to a monotonicity property of the degree-coloring index.

Contribution

It defines degree-coloring, formulates a conjecture linking it to graph parameters, and establishes an equivalence condition for the conjecture's validity.

Findings

Conjecture relates degree-coloring index to maximum degree and density.

Proves the conjecture holds iff the index is a monotone function.

Establishes a new theoretical framework for multigraph coloring.

Abstract

A notion of degree-coloring is introduced; it captures some, but not all properties of standard edge-coloring. We conjecture that the smallest number of colors needed for degree-coloring of a multigraph $G$ [the degree-coloring index $\tau(G)$] equals $\max\{\Delta, \omega\}$, where $\Delta$ and $\omega$ are the maximum vertex degree in $G$ and the multigraph density, respectively. We prove that the conjecture holds iff $\tau(G)$ is a monotone function on the set of multigraphs.