On Interval Non-Edge-Colorable Eulerian Multigraphs

TL;DR

This paper proves that Eulerian multigraphs with an odd number of edges cannot have interval colorings and introduces methods to construct such non-colorable graphs.

Contribution

It establishes a fundamental non-colorability property for Eulerian multigraphs with odd edges and provides construction techniques for non-colorable cases.

Findings

Eulerian multigraphs with odd edges are not interval colorable

Methods for constructing interval non-edge-colorable Eulerian multigraphs

Clarifies limitations of interval edge-colorings in multigraphs

Abstract

An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.