The cubicity of hypercube graphs

For a graph G, its cubicity is the minimum dimension k such that G is representable as the intersection graph of (axis-parallel) cubes in k-dimensional space. (A k-dimensional cube is a Cartesian product R₁×R₂×?×R_k, where R_i is a closed interval of the form [a_i,a_i+1] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113-118] showed that for a d-dimensional hypercube H_d, . In this paper, we use the probabilistic method to show that . The parameter boxicity generalizes cubicity: the boxicity of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k-dimensional space. Since for any graph G, our result implies that . The problem of determining a non-trivial lower bound for is left open.     

Monotonicity of the minimum cardinality of an identifying code in the hypercube

Let M_t(n) denote the minimum cardinality of a t-identifying code in the n-cube. It was conjectured that for all n?2 and t?1 we have M_t(n)?M_t(n+1). We prove this inequality for t=1.     

On even-cycle-free subgraphs of the hypercube

It is shown that the size of any C4k+2-free subgraph of the hypercube Qn, k?3, is o(e(Qn)).     

Embedding complete trees into the hypercube

We consider embeddings of the complete t-ary trees of depth k (denotation T^k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T^k,t), is known if max{k,t}2. First, we study the next open cases t=3 and k=3. We improve the known upper bound dim(T^k,3)2k+1 up to lim_k→∞dim(T^k,3)/k5/3 and show lim_t→∞dim(T^3,t)/t=227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T^k,t) for arbitrary k and t.     

Exact hyperplane covers for subsets of the hypercube

Alon and Füredi (1993) showed that the number of hyperplanes required to cover ${\{0,1\}}^n?\{0\}$ without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering ${\{0,1\}}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.     

The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph G, denoted D(G), is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in Bogstad and Cowen [The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29-35] by showing that the distinguishing number of , the pth graph power of the n-dimensional hypercube, is 2 whenever 2<p<n-1. This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube AQ_n, a variant of the hypercube introduced in Choudum and Sunitha [Augmented cubes, Networks 40 (2002) 71-84]. We show that D(AQ₁)=2; D(AQ₂)=4; D(AQ₃)=3; and D(AQ_n)=2 for n?4. The sequence of distinguishing numbers answers a question raised in Albertson and Collins [An introduction to symmetry breaking in graphs, Graph Theory Notes N.Y. 30 (1996) 6-7].     

Vertex Turán problems in the hypercube

Let Qn be the n-dimensional hypercube: the graph with vertex set n{0,1} and edges between vertices that differ in exactly one coordinate. For 1?d?n and F⊆d{0,1} we say that S⊆n{0,1} is F-free if every embedding satisfies i(F)?S. We consider the question of how large S⊆n{0,1} can be if it is F-free. In particular we generalise the main prior result in this area, for F=²{0,1}, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal sets.We also show that the density required to guarantee an embedded copy of at least one of a family of forbidden configurations may be significantly lower than that required to ensure an embedded copy of any individual member of the family.Finally we show that any subset of the n-dimensional hypercube of positive density will contain exponentially many points from some embedded d-dimensional subcube if n is sufficiently large.     

The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim (1998) conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is $2^{n?2}$, for all $n\ge 2$. This was revised by Riddle (2002), who conjectured that it is at least $2^{n?2}$, and proved it for all even $n$. We show that the revised conjecture holds for all $n\ge 2$. The proof is based on simple linear algebra.     

A note on short cycles in a hypercube

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd?s about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C_l) be the largest number of edges in a subgraph of a hypercube Q_n containing no cycle of length l. It is known that f(n,C_l)=o(|E(Q_n)|), when l=4k, k?2 and that . It is an open question to determine f(n,C_l) for l=4k+2, k?2. Here, we give a general upper bound for f(n,C_l) when l=4k+2 and provide a coloring of E(Q_n) by four colors containing no induced monochromatic C₁₀.     

On maximal 3-restricted edge connectivity and reliability analysis of hypercube networks

It is shown in this work that all n-dimensional hypercube networks for n ? 4 are maximally 3-restricted edge connected. Employing this observation, we analyze the reliability of hypercube networks and determine the first 3n − 5 coefficients of the reliability polynomial of n-cube networks.     

Excessive near 1-factorizations

We begin the study of sets of near 1-factors of graphs G of odd order whose union contains all the edges of G and determine, for a few classes of graphs, the minimum number of near 1-factors in such sets.     

A hypercube queueing loss model with customer-dependent service rates

This paper is concerned with the solution of a specific hypercube queueing model. It extends the work that was described in a related paper by Atkinson et al. [Atkinson, J.B., Kovalenko, I.N., Kuznetsov, N., Mykhalevych, K.V., 2006. Heuristic methods for the analysis of a queuing system describing emergency medical services deployed along a highway. Cybernetics & Systems Analysis, 42, 379–391], which investigated a model for deploying emergency services along a highway. The model is based on the servicing of customer demands that arise in a number of distinct geographical zones, or atoms. Service is provided by servers that are positioned at a number of bases, each having a fixed geographical location along the highway. At each base a single server is available. Demands arising in any atom have a first-preference base and a second-preference base. If the first-preference base is busy, service is provided by the second-preference base; and, if both bases are busy, the demand is lost. In practice, because of differences in travel times from the first and second-preference bases to the atom in question, the service rate may be significantly different in the two cases. The model studied here allows for such customer-dependent service rates to occur, and the corresponding hypercube model has 3ⁿ states, where n is the number of bases. The computational intractability of this model means that exact solutions for the long-run proportion of lost demands (p_loss) can be obtained only for small values of n. In this paper, we propose two heuristic methods and a simulation approach for approximating p_loss. The heuristics are shown to produce very accurate estimates of p_loss.     

Some Turán type results on the hypercube

For graphs H,G a classical problem in extremal graph theory asks what proportion of the edges of H a subgraph may contain without containing a copy of G. We prove some new results in the case where H is a hypercube. We use a supersaturation technique of Erd?s and Simonivits to give a characterization of a set of graphs such that asymptotically the answer is the same when G is a member of this set and when G is a hypercube of some fixed dimension. We apply these results to a specific set of subgraphs of the hypercube called Fibonacci cubes. Additionally, we use a coloring argument to prove new asymptotic bounds on this problem for a different set of graphs. Finally we prove a new asymptotic bound for the case where G is the cube of dimension 3.     

Maximal sets of Hamilton cycles in Knr;λ1,λ2

Let $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ be the $r$-partite multigraph in which each part has size $n$, where two vertices in the same part or different parts are joined by exactly ${\lambda }_1$ edges or ${\lambda }_2$ edges, respectively. It is proved that there exists a maximal set of $t$ edge-disjoint Hamilton cycles in $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ for ${\lambda }_{2}n\lfloor \frac{r+3}{4}\rfloor \le t\le min\left\{\lfloor \frac{{\lambda }_2{n}^2(r-1)}{2}\rfloor ,\lfloor \frac{{\lambda }_1(n-1)+{\lambda }_{2}n(r-1)}{2}\rfloor \right\}$, the upper bound being best possible. The results proved make use of the method of amalgamations.     

A Ramsey‐type result for the hypercube

We prove that for every fixed k and ? ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Q_n with k colors contains a monochromatic cycle of length 2 ?. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, that is, have the property that for every k, any k‐edge coloring of a sufficiently large Q_n contains a monochromatic copy of H. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 196–208, 2006     

Approximate Hamilton decompositions of random graphs

We show that if pn ? log n the binomial random graph G_n,p has an approximate Hamilton decomposition. More precisely, we show that in this range G_n,p contains a set of edge‐disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn ? log n is necessary. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

None of the coronoid systems can be isometrically embedded into a hypercube

A graph that can be isometrically embedded into a hypercube is called a partial cube (or binary Hamming graph). Klavžar, Gutman and Mohar [S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relations, J. Chem. Inf. Comput. Sci. 35 (1995) 590–593] showed that all benzenoid systems are partial cubes. In this article we show that none of the coronoid systems (benzenoid systems with “holes”) is a partial cube.     

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let G_n,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property A_k, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000     

Hamilton cycles in circulant digraphs with prescribed number of distinct jumps

Let G be a digraph that consists of a finite set of vertices V(G). G is called a circulant digraph if its automorphism group contains a |V(G)|-cycle. A circulant digraph G has arcs incident to each vertex i, where integers a_ks satisfy 0<a₁<a₂<a_j≤|V(G)|−1 and are called jumps. It is well known that not every G is Hamiltonian. In this paper we give sufficient conditions for a G to have a Hamilton cycle with prescribed distinct jumps, and prove that such conditions are also necessary for every G with two distinct jumps. Finally, we derive several results for obtaining G^′ with k, k≥2 distinct jumps if the corresponding G contains a Hamilton cycle with two distinct jumps.