Generalised Mycielski graphs and the Borsuk-Ulam theorem

TL;DR

This paper demonstrates that the chromatic number of generalized Mycielski graphs can be characterized using a combinatorial approach based on Fan's lemma, establishing an equivalence with the Borsuk-Ulam theorem.

Contribution

It provides an elementary combinatorial proof of Stiebitz's theorem and shows its equivalence to the Borsuk-Ulam theorem, bridging topological and combinatorial methods.

Findings

Elementary combinatorial proof of Stiebitz's theorem

Equivalence between Stiebitz's theorem and the Borsuk-Ulam theorem

Connection established via Fan's combinatorial lemma

Abstract

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovasz, which invokes the Borsuk-Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk-Ulam theorem.