A family of multigraphs with large palette index

TL;DR

This paper introduces a new family of multigraphs with a palette index that can be expressed as a quadratic polynomial of the maximum degree, advancing understanding of edge-coloring properties.

Contribution

The paper defines the palette multigraph and constructs the first known multigraph family with palette index expressed quadratically in maximum degree.

Findings

Established properties of the palette multigraph.

Constructed multigraphs with palette index as a quadratic function.

Connected results to broader questions in edge-coloring theory.

Abstract

Given a proper edge-coloring of a loopless multigraph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. The palette index of a multigraph is defined as the minimum number of distinct palettes occurring among the vertices, taken over all proper edge-colorings of the multigraph itself. In this framework, the palette multigraph of an edge-colored multigraph is defined in this paper and some of its properties are investigated. We show that these properties can be applied in a natural way in order to produce the first known family of multigraphs whose palette index is expressed in terms of the maximum degree by a quadratic polynomial. We also attempt an analysis of our result in connection with some related questions.