On the number of star-shaped classes in optimal colorings of Kneser graphs

TL;DR

This paper investigates the structure of optimal colorings of Kneser graphs, specifically addressing the number of star-shaped classes, and provides a negative answer to a previously posed question for the case k=2.

Contribution

It offers a negative resolution to a question about star-shaped classes in optimal colorings of Kneser graphs for k=2, advancing understanding of their combinatorial properties.

Findings

Negative answer to the question for k=2

Insights into the structure of star-shaped classes in Kneser graphs

Contributions to the theory of graph colorings and combinatorics

Abstract

A family of sets is called star-shaped if all the members of the family have a point in common. The main aim of this paper is to provide a negative answer to the following question raised by James Aisenberg et al [Short proofs of the kneser-Lovasz coloring principle, Information and Computation, 261:296-310, 2018.], for the case k=2.