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.
Highlights
The Mycielski construction [10] is one of the earliest and arguably simplest constructions of triangle-free graphs of arbitrary chromatic number
It is an easy exercise to show that the chromatic number increases with each iteration of M2(·)
The construction was generalised by Stiebitz [15], and independently by Van Ngoc [16], in the following way
Summary
The Mycielski construction [10] is one of the earliest and arguably simplest constructions of triangle-free graphs of arbitrary chromatic number. Stiebitz [15] was able to show that the chromatic number does increase with each iteration of Mr(·) if we start with an odd cycle, or some other suitably chosen graph. At the end of their paper, Van Ngoc and Tuza [17] propose the following problem: we would like to invite attention to the problem that no elementary combinatorial proof is known so far for the general form of Stiebitz’s theorem, yielding graphs of arbitrarily large chromatic number and fairly large odd girth. We would like to point out that our proof of Theorem 1 leads to a discrete proof of Schrijver’s [13] sharpening of the Lovasz–Kneser theorem [7], via the following result of Kaiser and Stehlık [6] (whose proof is entirely combinatorial). For all integers k 1 and n > 2k, there exists a graph G ∈ Mn−2k+2 homomorphic to SG(n, k)
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