Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

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 edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs

The minimum cost flow problem with interval and fuzzy arc costs *

Abstract We follow the total order for intervals and fuzzy numbers introduced by Hashemi et al. in [1] and Ghatee et al. in [2] to solve the minimum cost flow problem with either, interval or fuzzy arc costs by using its crisp model, a minimum cost flow problem associated to original imprecise problem. Numerical simulations compare the performance of this method in real scenarios with the algorithm proposed in 2010 Mathematics Subject Classification: 65G30, 65G40, 03E72, 90C70

Triangulations and a generalization of Bose's method

AbstractWe present a nontrivial extension to Bose's method for the construction of Steiner triple systems, generalizing the traditional use of commutative and idempotent quasigroups to employ a new algebraic structure called a 3-tri algebra. Links between Steiner triple systems and 2-(v,3,3) designs via 3-tri algebras are also explored

Two identification protocols based on Cayley graphs of Coxeter groups ∗

A challenge-response identification protocol is introduced, based on the intractability of the word problem in some Coxeter groups. A Prover builds his public key as the set of leaves of a tree in the Cayley graph of a Coxeter group, and the tree itself is his private keys. Any challenge posed by a Verifier consists of a subset of the public key, and the Prover shows his knowledge of the private key by providing a subtree having as set of leaves the challenge set. Any third party aiming to impersonate the Prover faces a form of the word problem in the Coxeter group. Although this protocol maintains the secrecy of the whole private key, it is disclosing some parts of it. A second protocol is introduced which is indeed a transcription of the already classical zero-knowledge protocol to recognize pairs of isomorphic graphs

Two discrete versions of the Inscribed Square Conjecture and some related problems

AbstractThe Inscribed Square Conjecture has been open since 1911. It states that any plane Jordan curve J contains four points on a non-degenerate square. In this article two different discrete versions of this conjecture are introduced and proved. The first version is in the field of digital topology: it is proved that the conjecture holds for digital simple closed 4-curves, and that it is false for 8-curves. The second one is in the topological graph theory field: it is proved that any cycle of the grid Z2 contains an inscribed square with integer vertices. The proofs are based on a theorem due to Pak. An infinite family of 4-curves in the digital plane containing a single non-degenerate inscribed square is introduced as well as a second infinite family containing one 4-curve with exactly n inscribed squares for each positive integer value of n. Finally an algorithm with time complexity O(n2) is given to find inscribed squares in simple digital curves

Selección aleatoria de árboles generadores en gráficas

Resumen: Existen diversos procedimientos para seleccionar aleatoriamente árboles generadores en gráficas conexas no dirigidas, con tiempos esperados de ejecución Entre los órdenes 0(n log n) y O(n3)en los peores casos, donde n es el número de vértices en la gráfica. En este trabajo realizamos la localización efectiva y eficiente de árboles generadores mediante paseos aleatorios sobre dichas gráficas, con la finalidad de obtener un equilibrio entre el diámetro del árbol, la valencia de los vértices internos y el número de hojas de los árboles obtenidos. Para esto, proponemos el uso de diversas matrices de transición en cadenas de Markov, considerando diferentes distribuciones de probabilidad para las vecindades de vértices involucradas en el paseo aleatorio