Homogeneous Edge-Colorings of Graphs

TL;DR

This paper introduces the concept of homogeneous edge-colorings in multigraphs, defines the homogeneous chromatic index, and determines this index for specific classes such as complete multigraphs, trees, and bipartite multigraphs.

Contribution

It formalizes the homogeneous edge-coloring concept and computes the homogeneous chromatic index for key classes of multigraphs, advancing graph coloring theory.

Findings

Determined hi(G) for complete multigraphs.

Calculated hi(G) for trees.

Established hi(G) for complete bipartite multigraphs.

Abstract

Let G = (V, E) be a multigraph without loops and for any x {\in}V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m >= 2 and a coloring c:E {\to} S of the edges of Gsuchthat|S| = mandforanyx{\in}V,if|E(x)| = mqx+rx with0 <= rx <m, there exists a partition of E(x) in rx color classes of cardinality qx + 1 and other m-rx color classes of cardinality qx. The homogeneous chromatic index \c{hi}(G) is the least m for which there exists such a coloring. We determine \c{hi}(G) in the case that G is a complete multigraph, a tree or a complete bipartite multigraph.