Generalised Mycielski graphs and the Borsuk-Ulam theorem

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.


Introduction
The Mycielski construction [10] is one of the earliest and arguably simplest constructions of triangle-free graphs of arbitrary chromatic number. Given a graph G = (V, E), we let M 2 (G) be the graph with vertex set V ×{0, 1}∪{z}, where there is an edge {(u, 0), (v, 0)} and {(u, 0), (v, 1)} whenever {u, v} ∈ E, and an edge {(u, 1), z} for all u ∈ V . It is an easy exercise to show that the chromatic number increases with each iteration of M 2 (·).
If r > 2, it is no longer true that the chromatic number increases with each iteration of M r (·). For instance, it can be shown that if C 7 is the complement of the 7-cycle, then χ(M 3 (C 7 )) = χ(C 7 ) = 4. However, Stiebitz [15] was able to show that the chromatic does increase with each iteration of M r (·) if we start with an odd cycle, or some other suitably chosen graph. For every integer k ≥ 2, let us denote by M k the set of all 'generalised Mycielski graphs' obtained from K 2 by k − 2 iterations of M r (·), where the value of r The first author was partially supported by NWO grants 639.032.529 and 612.001.409. Part of this work was carried out while this author visited Laboratoire G-SCOP supported by ANR Project Stint (ANR-13-BS02-0007) and LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).
Stiebitz's proof is based on Lovász's [7] bound on the chromatic number in terms of the connectedness of the neighbourhood complex, which Lovász developed to prove Kneser's conjecture (see [8] for a comprehensive account). Lovász's bound uses the following result of Borsuk [2], usually known in the literature as the Borsuk-Ulam theorem. Theorem 1.2 (Borsuk [2]). There exists no continuous antipodal mapping f : S n → S n−1 ; that is, a continuous mapping such that To this day, no combinatorial proof of Theorem 1.1 is known (see [8, pp. 133]), except for the case k = 4 [17]. At the end of their paper, Van Ngoc and Tuza [17] propose the following problem: Finally, 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.
The answer to the problem depends on the interpretation of 'elementary combinatorial proof'. Does it mean a proof that is 'discrete' and does not rely on continuity? Or does it mean a 'graph theoretic' proof which avoids any topological concepts, such as triangulations of spheres?
In this note we will give a new discrete proof of Theorem 1.1 based on a generalisation, due to Prescott and Su [11], of a classical lemma of Fan [4], and on a result of Kaiser and Stehlík [6]. Since the proofs of both these theorems are discrete, this provides a discrete proof of Theorem 1.1.
Triangulations of spheres are central to our proof, and rewriting the proof so as to avoid any topological concepts (as Matoušek [9] has done for the Lovász-Kneser theorem) is certainly possible, but seems somewhat artificial. Indeed, we show that Theorem 1.2 follows fairly easily from Theorem 1.1.
We would like to point out that our proof of Theorem 1.1 leads to a new proof of Schrijver's [13] sharpening of the Lovász-Kneser theorem [7], via the following result of Kaiser and Stehlík [6] (whose proof is entirely combinatorial). For a definition of SG(n, k), we refer the reader to [6] or [8]. [6]). For all integers k ≥ 1 and n > 2k, there exists a graph G ∈ M k homomorphic to SG(n, k).

Preliminaries
Our graph theoretic terminology is standard and follows [1]. For an excellent introduction to topological methods in combinatorics, and all the topological terms used in this paper, see [8].
Prescott and Su [11] introduced flags of hemispheres to prove a slight generalisation of Fan's combinatorial lemma [4]. A flag of hemispheres in S n is a sequence H 0 ⊂ · · · ⊂ H n where each H d is homeomorphic to a d-ball, {H 0 , −H 0 } are antipodal points, H n ∪ −H n = S n , and for 1 ≤ d ≤ n, The polyhedron |K| of a simplicial complex K is defined as the union of all of its simplices. We say that K is a triangulation of |K| (or any space homeomorphic to it). A triangulation K of S n is (centrally or antipodally) symmetric if σ ∈ K whenever −σ ∈ K. A symmetric triangulation K of S n is said to be aligned with hemispheres if we can find a flag of hemispheres such that for every d, there is a subcomplex of the d-skeleton of K that triangulates H d .
Given a simplicial complex K and a labelling (map) λ : [4] is a key ingredient of our proof.
Theorem 2.1 (Prescott and Su [11]). Let K be a symmetric triangulation of S n aligned with hemispheres, and let λ : Then there exists an odd number of positive alternating n-simplices. In particular, k ≥ n + 1.
We remark that the proof in [11] is constructive and discrete, and that Fan's original result [4] imposes a more restrictive condition on the triangulation.
Suppose K is a symmetric triangulation of S n . A 2-colouring of K is an assignment of two colours (black and white) to the vertices of K. The 2-colouring is said to be antisymmetric if antipodal vertices receive distinct colours, and it is proper if no n-simplex is monochromatic.
Given a symmetric triangulation K of S n and a proper antisymmetric 2-colouring κ of K, we denote byG(K, κ) the graph obtained from the 1-skeleton K (1) by deleting all monochromatic edges. If ν denotes the antipodal action onG(K, κ), we set G(K, κ) =G(K, κ)/ν, and let p :G(K, κ) → G(K, κ) be the corresponding projection. Note that the graphG(K, κ) is a bipartite double cover of G(K, κ).
The following theorem is an immediate consequence of [6, Lemma 3.2 and Theorem 6.1], where the results are stated in terms of so-called quadrangulations of projective spaces.

A combinatorial proof of Theorem 1.1
Our proof of Theorem 1.1 is based on the following corollary of Theorem 2.1.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. The case k = 2 (G = K 2 ) and k = 3 (G is an odd cylce) are trivial, so assume k > 3 and let G ∈ M k . The graph G is obtained from an odd cycle by k − 3 iterations of M r (·), where the value of r can vary from iteration to iteration. By repeated applications of Theorem 2.2 (k − 3 applications to be exact), there exists a symmetric triangulation K of S k−2 aligned with hemispheres, and a proper antisymmetric 2-colouring κ such that G ∼ = G(K, κ). (To see this, observe that M r (K 2 ) is isomorphic to the odd cycle C 2r+1 , which is isomorphic to G(K, κ), where K is a symmetric triangulation of S 1 -i.e., a graph-isomorphic to the cycle C 4r+2 , and κ is a proper 2-colouring of K. By choosing any pair of antipodal vertices of K to be the hemispheres H 0 and −H 0 , it is clear that K is aligned with hemispheres.) Let us say the colours used in κ are black and white.

Equivalence of the theorems of Borsuk-Ulam and Stiebitz
Let us recall the following construction due to Erdős and Hajnal [3]. The Borsuk graph BG(n, α) is defined as the (infinite) graph whose vertices are the points of R n+1 on S n , and the edges connect points at Euclidean distance at least α, where 0 < α < 2. Using Theorem 1.2, it can be shown that χ(G) ≥ n + 2 (in fact the two statements are equivalent, as noted by Lovász [7]). Furthermore, by using the standard (n + 2)-colouring of S n based on the central projection of a regular (n + 1)-simplex, it can be shown that BG(n, α) is (n + 2)-chromatic for all α sufficiently large. In particular, Simonyi and Tardos [14] have shown that BG(n, α) is (n + 2)-chromatic for all α ≥ α 0 , where α 0 = 2 1 − 1/(n + 3).
We will now show how Theorem 1.2 can be deduced from Theorem 1.1 and Lemma 4.1.
Proof of Theorem 1.2. Suppose there exists a continuous antipodal map f : S n → S n−1 . Set ε = 1/ √ n + 2. Since every continuous function on a compact set is uniformly continuous, there exists δ > 0 such that if x−y < δ, then f (x) − f (y) < 2ε.