http://hdl.handle.net/2117/3178 2013-06-19T14:31:40Z http://hdl.handle.net/2117/19540 Title: Partitions of graphs into small and large sets Authors: Bojilov, Asen; Caro, Yair; Hansberg Pastor, Adriana; Nedyalko, Nevno Abstract: Let GG be a graph on nn vertices. We call a subset AA of the vertex set V(G)V(G)kk-small if, for every vertex v∈Av∈A, deg(v)≤n−|A|+kdeg(v)≤n−|A|+k. A subset B⊆V(G)B⊆V(G) is called kk-large if, for every vertex u∈Bu∈B, deg(u)≥|B|−k−1deg(u)≥|B|−k−1. Moreover, we denote by φk(G)φk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-small sets, and by Ωk(G)Ωk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-large sets. In this paper, we will show tight connections between kk-small sets, respectively kk-large sets, and the kk-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both φk(G)φk(G) and Ωk(G)Ωk(G) and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro–Wei Theorem, Turán’s Theorem and the Hansen–Zheng Theorem among other things. 2013-06-12T13:07:19Z http://hdl.handle.net/2117/19517 Title: The efficiency of tissue P systems with cell separation relies on the environment Authors: Macias Ramos, Luis F.; Pérez Jiménez, Mario J.; Riscos Nuñez, Agustín; Rius Font, Miquel; Valencia Cabrera, Luis Abstract: The classical definition of tissue P systems includes a distinguished alphabet with the special assumption that its elements are available in an arbitrarily large amount of copies. These objects are shared in a distinguished place of the system, called the environment. This ability of having infinitely many copies of some objects has been widely exploited in the design of efficient solutions to computationally hard problems by means of tissue P systems. This paper deals with computational aspects of tissue P systems with cell separation where there is no such environment as described above. The main result is that only tractable problems can be efficiently solved by using this kind of P systems. Bearing in mind that NP–complete problems can be efficiently solved by using tissue P systems without environment and with cell division, we deduce that in the framework of tissue P systems without environment, the kind of rules (separation versus division) provides a new frontier of the tractability of decision problems. 2013-06-05T12:21:28Z http://hdl.handle.net/2117/19434 Title: On k-domination and j-independence in graphs Authors: Hansberg Pastor, Adriana; Pepper, Ryan Abstract: Let GG be a graph and let kk and jj be positive integers. A subset DD of the vertex set of GG is a kk-dominating set if every vertex not in DD has at least kk neighbors in DD. The kk-domination numberγk(G)γk(G) is the cardinality of a smallest kk-dominating set of GG. A subset I⊆V(G)I⊆V(G) is a jj-independent set of GG if every vertex in II has at most j−1j−1 neighbors in II. The jj-independence numberαj(G)αj(G) is the cardinality of a largest jj-independent set of GG. In this work, we study the interaction between γk(G)γk(G) and αj(G)αj(G) in a graph GG. Hereby, we generalize some known inequalities concerning these parameters and put into relation different known and new bounds on kk-domination and jj-independence. Finally, we will discuss several consequences that follow from the given relations, while always highlighting the symmetry that exists between these two graph invariants. 2013-05-29T07:33:27Z http://hdl.handle.net/2117/19287 Title: The Manhattan product of digraphs Authors: Comellas Padró, Francesc de Paula; Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel Abstract: We study the main properties of a new product of bipartite digraphs which we call Manhattan product. This product allows us to understand the subjacent product in the Manhattan street networks and can be used to built other networks with similar good properties. It is shown that if all the factors of such a product are (directed) cycles, then the digraph obtained is a Manhattan street network, a widely studied topology for modeling some interconnection networks. To this respect, it is proved that many properties of these networks, such as high symmetries, reduced diameter and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs. Moreover, we show that the Manhattan product of two Manhattan streets networks is also a Manhattan street network. Finally, some sufficient conditions for the Manhattan product of two Cayley digraphs to be also a Cayley digraph are given. Throughout our study we use some interesting recent concepts, such as the unilateral distance and related graph invariants. 2013-05-16T11:26:48Z http://hdl.handle.net/2117/19014 Title: Aula virtual de soporte a la docencia y al autoaprendizaje del cálculo para estudiantes universitarios con material docente y aplicaciones basadas en software libre Authors: Jarauta Bragulat, Eusebio; Pelayo Melero, Ignacio Manuel 2013-04-26T12:40:48Z http://hdl.handle.net/2117/19012 Title: Aula virtual de apoyo a la docencia y al autoaprendizaje de cálculo para estudiantes universitarios con material docente y aplicaciones basadas en software libre Authors: Jarauta Bragulat, Eusebio; Pelayo Melero, Ignacio Manuel 2013-04-26T12:30:02Z http://hdl.handle.net/2117/18915 Title: On the restricted connectivity and superconnectivity in graphs with given girth Authors: Balbuena Martínez, Maria Camino Teófila; Cera, M; Diánez, A; García-Vázquez, P; Marcote Ordax, Francisco Javier Abstract: The restricted connectivity κ′(G)κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex uu has all its neighbors in X; the superconnectivity κ1(G)κ1(G) is defined similarly, this time considering only vertices uu in G-XG-X, hence κ1(G)⩽κ′(G)κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u)d(u) standing for the degree of a vertex uu. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(G)κ1(G)⩾ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17–31] and guaranteeing κ1(G)=κ′(G)=ξ(G)κ1(G)=κ′(G)=ξ(G) under some additional constraints. Description: "Discrete Mathemetics Top Cited Article 2005-2010" 2013-04-22T11:28:54Z http://hdl.handle.net/2117/18912 Title: Moments in graphs Authors: Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest Abstract: Let G be a connected graph with vertex set V and a weight function that assigns a nonnegative number to each of its vertices. Then, the -moment of G at vertex u is de ned to be M G(u) = P v2V (v) dist(u; v), where dist( ; ) stands for the distance function. Adding up all these numbers, we obtain the -moment of G: This parameter generalizes, or it is closely related to, some well-known graph invari- ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the degree distance D0(G), obtained when (u) = (u), the degree of vertex u. In this paper we derive some exact formulas for computing the -moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding -moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same -moment for every (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product. 2013-04-22T10:41:17Z http://hdl.handle.net/2117/18908 Title: Polynomials in finite geometries Authors: Ball, Simeon Michael Abstract: A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs and unitals. In the case of nuclei these methods give lower bounds on the number of nuclei to a set of points in PG(n, q), usually dependent on some binomial coefficient not vanishing modulo the characteristic of the field. These lower bounds on nuclei lead directly to lower bounds on affine blocking sets with respect to lines. A short description of how linear polynomials can be used to construct maximal arcs in certain translation planes is included. A proof of the non-existence of maximal arcs in PG(2, q) when q is odd is outlined and some bounds are given as to when a (k, n)-arc can be extended to a maximal arc in PG(2, q). These methods can also be applied to unitals embedded in PG(2, q). One implication of this is that when q is the square of a prime a non-classical unital has a limited amount of Baer sublines amongst its secants. 2013-04-22T08:26:51Z http://hdl.handle.net/2117/18893 Title: Symplectic spreads and permutation polynomials Authors: Ball, Simeon Michael; Zieve, Michael Abstract: Every symplectic spread of PG(3, q), or equivalently every ovoid of Q(4, q), is shown to give a certain family of permutation polynomials of GF(q) and vice-versa. This leads to an algebraic proof of the existence of the Tits-L¨uneburg spread of W(22h+1) and the Ree-Tits spread of W(32h+1), as well as to a new family of low-degree permutation polynomials over GF(32h+1). 2013-04-19T12:30:37Z http://hdl.handle.net/2117/18891 Title: Commutative semifields of rank 2 over their middle nucleus Authors: Ball, Simeon Michael; Lavrauw, Michel Abstract: This article is about finite commutative semifields that are of rank 2 over their middle nucleus, the largest subset of elements that is a finite field. These semifields have a direct correspondence to certain flocks of the quadratic cone in PG(3, q) and to certain ovoids of the parabolic space Q(4, q). We shall consider these links, the known examples and non-existence results. 2013-04-19T12:12:41Z http://hdl.handle.net/2117/18848 Title: Polynomials in finite geometry Authors: Ball, Simeon Michael; Blokhuis, Aart 2013-04-17T10:13:03Z http://hdl.handle.net/2117/18846 Title: The polynomial method in Galois geometries Authors: Ball, Simeon Michael Abstract: The polynomial method refers to the application of polynomials to combinatorial problems. The method is particularly effective for Galois geometries and a number of problems and conjectures have been solved using the polynomial method. In many cases the polynomial approach is the only method which we know of that works. In this article, the various polynomial techniques that have been applied to Galois geometries are detailed and, to demonstrate how to apply these techniques, some of the problems referred to above are resolved. 2013-04-17T09:56:10Z http://hdl.handle.net/2117/18603 Title: New approach to the k-independence number of a graph Authors: Caro, Yair; Hansberg Pastor, Adriana Abstract: Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With k(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, k(G) > k+1 dde+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107]. 2013-04-04T10:54:16Z http://hdl.handle.net/2117/18452 Title: Clasificación de 3-semigrupos numéricos mediante L-formas Authors: Aguiló Gost, Francisco de Asis L.; Marijuan López, Carlos Abstract: Un semigrupo num erico generado por tres elementos tiene asociadas una o dos teselaciones peri odicas del plano generadas por una baldosa en forma de L. Se conocen algunas propiedades combinatorias del semigrupo en t erminos de las di- mensiones de dicha baldosa. Por otro lado, algunas clases de semigrupos num ericos se han utilizado en el estudio de singularidades de curvas, donde reciben el nombre de las clases de curvas asociadas. En este trabajo damos una caracterizaci on de las dimensiones de las baldosas asocidas a un 3-semigrupo num erico en t erminos de sus generadores y la utilizamos para clasi car los 3-semigrupos num ericos de inter es en geometr a. 2013-03-21T12:49:42Z