With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph. In particular, the set of satisfiable formulae in k-conjunctive normal form with ≤ n variables has the homotopy type of Θ(Cube(n, n− k)), where Cube(n, n− k) is a hypergraph associated to the (n− k)-skeleton of an n-cube. We make...

We prove the NP-completeness of finding a Hamiltonian path in an N ×N ×N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N ×N ×N cube. Along the way, we prove a universality result that zig-zag chains...

We prove the NP-completeness of finding a Hamiltonian path in an N ×N ×N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N ×N ×N cube. Along the way, we prove a universality result that zig-zag chains...

The cube graph Qn is the skeleton of the n-dimensional cube. It is an n-regular graph on 2 n vertices. The Ramsey number r(Qn,Ks) is the minimum N such that every graph of order N contains the cube graph Qn or an independent set of order s. In 1983, Burr and Erdős asked whether the simple lower bound r(Qn,Ks) ≥ (s − 1)(2n − 1) + 1 is tight for s fixed and n sufficiently large. We make progress ...

A unique sink orientation (USO) is an orientation of the n-dimensional cube graph (n-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is nΘ(2 n) [13]. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn...

We show that by exchanging any two independent edges in any shortest cycle of the n-cube (n 2 3), its diameter decreases by one unit. This leads us to define a new class of n-regular graphs, denoted Ten, with Zn vertices and diameter n 1, which has the (n l)-cube as subgraph. Other properties of TQ,, such as connectivity and the lengths of the disjoints paths are also investigated. Moreover, we...

In the present paper we find a bijection between the set of small covers over an n-cube and the set of acyclic digraphs with n labeled nodes. Using this, we give a formula of the number of small covers over an n-cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and Zn2 -equivariant diffeomorphism classes. Moreover we prove that the number of acyclic digraphs...

A closed knight’s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. When the chessboard is translated into graph theoretic terms the question is transformed into the existence of a Hamiltonian cycle. There are two common tours to consider on the cube. One is to tour the six exterior n × n boards that form the cube. The ot...

A slight modification of the proof of Szemerédi’s cube lemma gives that if a set S ⊂ [1, n] satisfies |S| ≥ n2 , then S must contain a non-degenerate Hilbert cube of dimension blog2 log2 n− 3c. In this paper we prove that in a random set S determined by Pr{s ∈ S} = 1 2 for 1 ≤ s ≤ n, the maximal dimension of nondegenerate Hilbert cubes is a.e. nearly log2 log2 n+log2 log2 log2 n and determine t...