Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Abstract

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, 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.

Edge-critical subgraphs of Schrijver graphs

“…Using topological methods, Stiebitz [14] (see also [2,9]) proved the following result. A 'discrete' proof, based on a combinatorial lemma of Fan, can be found in [10].…”

Edge-critical subgraphs of Schrijver graphs II: The general case

“…Using topological methods, Stiebitz [14] (see also [2,9]) proved the following result. A 'discrete' proof, based on a combinatorial lemma of Fan, can be found in [10]. We now come to the key lemma of this section.…”

A Note on the Immersion Number of Generalized Mycielski Graphs

“…The notation in Definition 3 again follows [6]. Notice that µ 1 (G) is obtained from a copy of G with an additional vertex adjacent to all vertices in G and µ 2 (G) is the Mycielskian of G, that is µ(G) = µ 2 (G).…”

“…The general construction was also described as the "cone over G" by Tardif [10]. While the chromatic number of the generalized Mycielski graphs does not necessarily increase when the construction is applied [6], Stiebitz [9] proved the chromatic number does increase if the construction is applied to a specific class of graphs. We will show that the immersion number of Mycielski graphs and generalized Mycielski graphs increases by at least 1, showing that these graphs behave as expected with respect to the Abu-Khzam-Langston Conjecture.…”