What is the minimal closed cone containing all f -vectors of cubical d-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold. R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vectorDenote by e i the i -th unit vector in R ⌊d /2⌋ . Stacked cubical polytopes show that the ray spanned by e 1 is in C d , and neighborly cubical polytopes show that the ray spanned by e ⌊d /2⌋ is in C d . Our main result is that all the rays spanned by the vectors e i are in C d . ThusThe conjecture of Adin and Babson-Billera-Chan isAn analogue of Theorem 1 was previously known for the much wider class of PL cubical spheres [BBC97, Theorem 5.7]. Also, Conjecture 2 holds for d ≤ 5, by combining the constructions above with Steve Klee's result [Kle11, Prop.3.7] asserting that g c k (Q) ≥ 0 for any cubical polytope Q of dimension 2k + 1. Sanyal and Ziegler [SZ10] showed how to construct, from any simplicial (d − 2)-polytope P on n − 1 vertices and a total order v 1 < v 2 < . . . < v n−1 on its vertices, a cubical d-polytope Q = Q(P, <) on 2 n vertices; it is the projection of a deformed n-cube in R n onto the last d coordinates. Further, they showed that if P is k-neighborly then the k-skeleton of Q is isomorphic to the k-skeleton of the n-cube. We apply their construction to the case where P n is the k-neighborly k-stacked (d − 2)-polytope on n − 1 vertices constructed by McMullen and Walkup [MW71], with 1 ≤ k ≤ ⌊ d −2 2 ⌋, and with a suitable total order <. Analyzing the cubical g-vectors of the resulting polytopes Q(k, d, n) = Q(P n , <), as n tends to infinity, gives Theorem 1. See Theorem 13 and Corollary 14 for the exact values and asymptotic behavior of the cubical g-vectors.The generalized lower bound theorem for simplicial polytopes (GLBT), conjectured by McMullen and Walkup [MW71] and proved by Murai and Nevo [MN13], asserts that for 1 ≤ k < ⌊ d 2 ⌋, a simplicial dpolytope P is k-stacked if and only if g k+1 (P ) = 0. The polytopes Q(k, d, n) demonstrate that the natural cubical analogue of the GLBT does not hold:Theorem 3. For any k ≥ 1 and n ≥ d ≥ 2k + 4, we have g c k+2 (Q(k, d, n)) = 0, and Q(k, d, n) is not cubical (k + 1)-stacked.The paper is organized as follows. Basic definitions and notation are given in section 2. In section 3 we define our variant of the McMullen-Walkup polytopes. A sketch of the Sanyal-Ziegler construction is given in section 4. In section 5 we construct the polytopes Q(k, d, n) mentioned above and analyze their cubical g-vector. In section 6 we prove Theorem 3, showing that Q(k, d, n) is not (k + 1)-stacked. The final section 7 concludes with remarks and open questions.

We introduce the notion of a t-graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented.

We consider the realization space of the d-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical d-polytopes, and apply this construction to certain cubical d-polytopes to conclude that the rays spanned by f -vectors of cubical d-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.

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We consider a generalization of the notion of partially balanced incomplete block designs (PBIBDs), by relaxing the requirement that the underlying association scheme be commutative. An infinite family of such generalizations is constructed, one for each prime power q congruent to 1 modulo 4.