COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
http://hdl.handle.net/2117/3178
Mon, 23 Oct 2017 17:31:34 GMT2017-10-23T17:31:34ZA finite version of the Kakeya problem
http://hdl.handle.net/2117/108967
A finite version of the Kakeya problem
Ball, Simeon Michael; Blokhuis, Aart; Domenzain, Diego
Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2(\frac{1}{2}N)^n$ plus smaller order terms, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a lower bound on the size of $S$ dependent on the ideal generated by the homogeneous polynomials vanishing on $D$. This bound is maximised as $(\frac{1}{2}N)^n$ plus smaller order terms, for $n\geqslant 4$, when $D$ contains the points of a $N^{n-1}$ grid.
Mon, 23 Oct 2017 11:15:14 GMThttp://hdl.handle.net/2117/1089672017-10-23T11:15:14ZBall, Simeon MichaelBlokhuis, AartDomenzain, DiegoLet $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2(\frac{1}{2}N)^n$ plus smaller order terms, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a lower bound on the size of $S$ dependent on the ideal generated by the homogeneous polynomials vanishing on $D$. This bound is maximised as $(\frac{1}{2}N)^n$ plus smaller order terms, for $n\geqslant 4$, when $D$ contains the points of a $N^{n-1}$ grid.Characterizing identifying codes through the spectrum of a graph or digraph
http://hdl.handle.net/2117/108920
Characterizing identifying codes through the spectrum of a graph or digraph
Balbuena Martínez, Maria Camino Teófila; Dalfó Simó, Cristina; Martínez Barona, Berenice
Fri, 20 Oct 2017 12:13:09 GMThttp://hdl.handle.net/2117/1089202017-10-20T12:13:09ZBalbuena Martínez, Maria Camino TeófilaDalfó Simó, CristinaMartínez Barona, BereniceColoración de grafos por vecindades diferentes.
http://hdl.handle.net/2117/108781
Coloración de grafos por vecindades diferentes.
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Alcom, Liliana; Gutierrez, Marisa
En este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.
Wed, 18 Oct 2017 07:07:14 GMThttp://hdl.handle.net/2117/1087812017-10-18T07:07:14ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelAlcom, LilianaGutierrez, MarisaEn este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.Limited broadcast domination: upper bounds and complexity
http://hdl.handle.net/2117/108780
Limited broadcast domination: upper bounds and complexity
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Cáceres, José; Puertas, M. Luz
En este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.
Wed, 18 Oct 2017 06:35:07 GMThttp://hdl.handle.net/2117/1087802017-10-18T06:35:07ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelCáceres, JoséPuertas, M. LuzEn este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.Locating domination in bipartites graphs and their complements
http://hdl.handle.net/2117/108779
Locating domination in bipartites graphs and their complements
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
En este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.
Wed, 18 Oct 2017 06:08:39 GMThttp://hdl.handle.net/2117/1087792017-10-18T06:08:39ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelEn este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas.On quotient digraphs and voltage digraphs
http://hdl.handle.net/2117/108606
On quotient digraphs and voltage digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Miller, Mirka; Ryan, Joe
We study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics,or to obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called a voltage digraph or base, with the spectrum of its lifted digraph Γα. We prove that all the eigenvalues of Γ (including multiplicities) are, in addition, eigenvalues of Γα. This study is carried out by introducing several reduced matrix representations of Γα. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.
Tue, 10 Oct 2017 15:03:49 GMThttp://hdl.handle.net/2117/1086062017-10-10T15:03:49ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelMiller, MirkaRyan, JoeWe study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics,or to obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called a voltage digraph or base, with the spectrum of its lifted digraph Γα. We prove that all the eigenvalues of Γ (including multiplicities) are, in addition, eigenvalues of Γα. This study is carried out by introducing several reduced matrix representations of Γα. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.From expanded digraphs to voltage and line digraphs
http://hdl.handle.net/2117/108605
From expanded digraphs to voltage and line digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Miller, Mirka; Ryan, Joe
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line digraphs, which are two known ways to obtain dense digraphs. In the same context, we show the equivalence between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, the De Bruijn and Kautz digraphs, where it is shown that both families can be seen as lifts of smaller De Bruijn digraphs with appropriate voltage assignments.
Tue, 10 Oct 2017 14:54:04 GMThttp://hdl.handle.net/2117/1086052017-10-10T14:54:04ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelMiller, MirkaRyan, JoeIn this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line digraphs, which are two known ways to obtain dense digraphs. In the same context, we show the equivalence between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, the De Bruijn and Kautz digraphs, where it is shown that both families can be seen as lifts of smaller De Bruijn digraphs with appropriate voltage assignments.The spectra of subKautz and cyclic Kautz digraphs
http://hdl.handle.net/2117/108455
The spectra of subKautz and cyclic Kautz digraphs
Dalfó Simó, Cristina
Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,l) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d=3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.
Fri, 06 Oct 2017 12:21:30 GMThttp://hdl.handle.net/2117/1084552017-10-06T12:21:30ZDalfó Simó, CristinaKautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,l) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d=3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.From subKautz digraphs to cyclic Kautz digraphs
http://hdl.handle.net/2117/105173
From subKautz digraphs to cyclic Kautz digraphs
Dalfó Simó, Cristina
Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)
Tue, 06 Jun 2017 12:12:17 GMThttp://hdl.handle.net/2117/1051732017-06-06T12:12:17ZDalfó Simó, CristinaKautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)On quotient digraphs and voltage digraphs
http://hdl.handle.net/2117/105171
On quotient digraphs and voltage digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Miller, Mirka; Ryan, Joe
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.
Tue, 06 Jun 2017 12:00:49 GMThttp://hdl.handle.net/2117/1051712017-06-06T12:00:49ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelMiller, MirkaRyan, JoeIn this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.