A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d 7. Our approach is inductive: for each fixed (d − 1)-dimensional 2-level polytope P 0 , we enumerate all d-dimensional 2-level polytopes P that have P 0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P 0 , we obtain all 2-level polytopes in dimension d.(1,3,4,5) Université libre de Bruxelles, Brussels, Belgium Lemma 4. Let P be a d-polytope having facet-defining inequalities g 1 (x) 0, . . . , g m (x) 0, and vertices v 1 , . . . , v n . If σ denotes a map from the affine hull aff(P ) of P to R m defined by σ(x) i := g i (x) for all x ∈ aff(P ), then the polytopes P and σ(P ) are affinely equivalent.Proof. The map σ : aff(P ) → R m is affine, and injective because it maps the vertices of any simplicial core for P to affinely independent points. The result follows.By definition, a polytope P is 2-level if and only if S(P ) can be scaled to be a 0/1 matrix. Given a 2-level polytope, we henceforth always assume that its facet-defining inequalities are scaled so that the slacks are 0/1. Thus, the slack embedding of a 2-level polytope depends only on the support 1 of its slack matrix, which only depends on its combinatorial structure. As a consequence, we have the following result:Lemma 5. Two 2-level polytopes are affinely equivalent if and only if they have the same combinatorial type.

Abstract. A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d 7. Our approach is inductive: for each fixed (d − 1)-dimensional 2-level polytope P 0 , we enumerate all d-dimensional 2-level polytopes P that have P 0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P 0 , we obtain all 2-level polytopes in dimension d.

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Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by showing that for any cubic integer α, there is a natural number n such that α + n is a chromatic root.

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\bv = \{v_e\}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over $K(\bv)$, and give three self-contained proofs that the Galois group of $\hat Z_M$ over $K(\bv)$ is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul