Bounds for the chromatic index of signed multigraphs

TL;DR

This paper extends classical edge-coloring theorems to signed multigraphs, establishing bounds on their chromatic index and characterizing those with minimal index.

Contribution

It introduces bounds for the chromatic index of signed multigraphs and characterizes balanced cases with minimal index, extending classical theorems to signed graphs.

Findings

Chromatic index of signed multigraphs is at most 1.5 times the maximum degree.

Balanced signed multigraphs have chromatic index at most maximum degree plus one.

Characterization of balanced signed multigraphs with minimal chromatic index.

Abstract

The paper studies edge-coloring of signed multigraphs and extends classical Theorems of Shannon and K\"onig to signed multigraphs. We prove that the chromatic index of a signed multigraph $(G,\sigma_G)$ is at most $\lfloor \frac{3}{2} \Delta(G) \rfloor$. Furthermore, the chromatic index of a balanced signed multigraph $(H,\sigma_H)$ is at most $\Delta(H) + 1$ and the balanced signed multigraphs with chromatic index $\Delta(H)$ are characterized.