Related papers: Two results about the hypercube

Let $\mathcal{A}$ be a union-closed family of sets with universe $\bigcup_{A \in \mathcal{A}}A = [n] = \{1,\cdots,n\}$ and length $\ell$. We prove that $|\mathcal{A}| \leq \sum_{i=0}^{\ell} \binom{n}{i}$, with equality if and only if…

Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on…

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…

A celebrated result of Alon and F\"{u}redi gives a tight lower bound on the number of hyperplanes required to cover all points of the Boolean cube $B^n$ except the origin $\bm{0}$. Recent breakthroughs by Sauermann and Wigderson generalized…

This paper focuses on the embeddability of hypercubes in an important class of Cayley graphs, known as augmented cubes. An $n$-dimensional augmented cube $AQ_n$ is constructed by augmenting the $n$-dimensional hypercube $Q_n$ with…

We consider two types of problems: maximising, over subsets $S\subseteq \{0,1\}^n$, the density of $d$-subcubes $C$ in the $n$-hypercube graph that span a subgraph such that $S\cap C$ is i) isomorphic to the given configuration…

Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the…

Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if…

For every irrational real $\alpha$, let $M(\alpha) = \sup_{n\geq 1} a_n(\alpha)$ denote the largest partial quotient in its continued fraction expansion (or $\infty$, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as…

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…

The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length…

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\ldots,n\}$, and let $e(H)$ denote the number of edges of $H$. Let $\nu(H)$ and $\tau(H)$…

Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the vector $\boldsymbol{x}=(x_1,\ldots, x_n) \in \mathbb{N}^n$ such that $M$ contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is a perfect…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

This paper introduces a new variant of hypercubes, which we call Z-cubes. The n-dimensional Z-cube $H_n$ is obtained from two copies of the (n-1)-dimensional Z-cube $H_{n-1}$ by adding a special perfect matching between the vertices of…

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $\lambda_n$. Using a novel construction of subcomponents we study…

Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case…

Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but…

Let F be a family of positive homothets (or translates) of a given convex body K in R^n. We investigate two approaches to measuring the complexity of F. First, we find an upper bound on the transversal number $\tau(F)$ of F in terms of $n$…

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight…