COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
http://hdl.handle.net/2117/3178
2020-02-18T13:46:21ZAn analogue of Vosper's theorem for extension fields
http://hdl.handle.net/2117/115387
An analogue of Vosper's theorem for extension fields
Bachoc, Christine; Serra Albó, Oriol; Zemor, Gilles
We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which \begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath} when dim FS, dim FT ¿ 2 and dim FST ¿ [L : F] - 2.
2018-03-19T10:51:16ZBachoc, ChristineSerra Albó, OriolZemor, GillesWe are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which \begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath} when dim FS, dim FT ¿ 2 and dim FST ¿ [L : F] - 2.Combinatorics in the Art of the Twentieth Century
http://hdl.handle.net/2117/115282
Combinatorics in the Art of the Twentieth Century
Barrière Figueroa, Eulalia
This paper is motivated by a question I asked myself: How can combinatorial structures be used in a work of art? Immediately, other questions arose: Whether there are artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between art works produced using
combinatorics? This paper presents the first results of the attempt to answer these questions, being a survey of a selection of works that use or contain combinatorics in some way, including music, literature and visual arts, focusing on the twentieth century.
2018-03-16T11:52:11ZBarrière Figueroa, EulaliaThis paper is motivated by a question I asked myself: How can combinatorial structures be used in a work of art? Immediately, other questions arose: Whether there are artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between art works produced using
combinatorics? This paper presents the first results of the attempt to answer these questions, being a survey of a selection of works that use or contain combinatorics in some way, including music, literature and visual arts, focusing on the twentieth century.Forbidden subgraphs in the norm graph
http://hdl.handle.net/2117/115152
Forbidden subgraphs in the norm graph
Ball, Simeon Michael; Pepe, Valentina
We show that the norm graph with n vertices about View the MathML source edges, which contains no copy of the complete bipartite graph Kt,(t-1)!+1, does not contain a copy of Kt+1,(t-1)!-1.
2018-03-14T10:45:08ZBall, Simeon MichaelPepe, ValentinaWe show that the norm graph with n vertices about View the MathML source edges, which contains no copy of the complete bipartite graph Kt,(t-1)!+1, does not contain a copy of Kt+1,(t-1)!-1.On the relation between graph distance and Euclidean distance in random geometric graphs
http://hdl.handle.net/2117/115054
On the relation between graph distance and Euclidean distance in random geometric graphs
Díaz Cort, Josep; Dieter Wilhelm, Mitsche; Perarnau Llobet, Guillem; Pérez Giménez, Xavier
Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=¿(vlogn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).
2018-03-12T09:24:10ZDíaz Cort, JosepDieter Wilhelm, MitschePerarnau Llobet, GuillemPérez Giménez, XavierGiven any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=¿(vlogn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).Revisiting Kneser’s theorem for field extensions
http://hdl.handle.net/2117/114080
Revisiting Kneser’s theorem for field extensions
Bachoc, Christine; Serra Albó, Oriol; Zemor, Gilles
A Theorem of Hou, Leung and Xiang generalised Kneser’s addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou’s conjecture. This result is a consequence of a strengthening of Hou et al.’s theorem that is inspired by an addition theorem of Balandraud and is obtained by combinatorial methods transposed and adapted to the extension field setting.
2018-02-13T14:33:26ZBachoc, ChristineSerra Albó, OriolZemor, GillesA Theorem of Hou, Leung and Xiang generalised Kneser’s addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou’s conjecture. This result is a consequence of a strengthening of Hou et al.’s theorem that is inspired by an addition theorem of Balandraud and is obtained by combinatorial methods transposed and adapted to the extension field setting.Transformation and decomposition of clutters into matroids
http://hdl.handle.net/2117/112728
Transformation and decomposition of clutters into matroids
Martí Farré, Jaume; Mier Vinué, Anna de
A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.
2018-01-12T13:52:45ZMartí Farré, JaumeMier Vinué, Anna deA clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.Random subgraphs make identification affordable
http://hdl.handle.net/2117/112330
Random subgraphs make identification affordable
Foucaud, Florent; Perarnau, Guillem; Serra Albó, Oriol
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that for every large enough ¿¿, every graph GG on nn vertices with maximum degree ¿¿ and minimum degree d=clog¿d=clog¿¿, for some constant c>0c>0, contains a large spanning subgraph which admits an identifying code with size O(nlog¿d)O(nlog¿¿d). In particular, if d=T(n)d=T(n), then GG has a dense spanning subgraph with identifying code O(logn)O(log¿n), namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code.
2017-12-20T12:55:54ZFoucaud, FlorentPerarnau, GuillemSerra Albó, OriolAn identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that for every large enough ¿¿, every graph GG on nn vertices with maximum degree ¿¿ and minimum degree d=clog¿d=clog¿¿, for some constant c>0c>0, contains a large spanning subgraph which admits an identifying code with size O(nlog¿d)O(nlog¿¿d). In particular, if d=T(n)d=T(n), then GG has a dense spanning subgraph with identifying code O(logn)O(log¿n), namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code.On star forest ascending subgraph decomposition
http://hdl.handle.net/2117/112325
On star forest ascending subgraph decomposition
Aroca Farrerons, José María; Lladó Sánchez, Ana M.
The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1¿¿¿Hn such that Hi has i edges and it is isomorphic to a subgraph of Hi+1, i=1,…,n-1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies dk-i=n-ifor each0=i=k-1, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.
2017-12-20T12:04:25ZAroca Farrerons, José MaríaLladó Sánchez, Ana M.The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1¿¿¿Hn such that Hi has i edges and it is isomorphic to a subgraph of Hi+1, i=1,…,n-1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies dk-i=n-ifor each0=i=k-1, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.Approximate results for rainbow labelings
http://hdl.handle.net/2117/112321
Approximate results for rainbow labelings
Lladó Sánchez, Ana M.; Miller, Mirka
A simple graph G=(V,E) is said to be antimagic if there exists a bijection f:E¿[1,|E|] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f:V¿[1,|V|], such that ¿x,y¿V, ¿xi¿N(x)f(xi)¿¿xj¿N(y)f(xj). Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval [1,2n+m-4] and, for trees with k inner vertices, in the interval [1,m+k]. In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree ¿ in the interval [1,n+t(n-t)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n-4k]. In particular, all trees with n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.
The final publication is available at Springer via https://doi.org/10.1007/s10998-016-0151-2]
2017-12-20T11:35:12ZLladó Sánchez, Ana M.Miller, MirkaA simple graph G=(V,E) is said to be antimagic if there exists a bijection f:E¿[1,|E|] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f:V¿[1,|V|], such that ¿x,y¿V, ¿xi¿N(x)f(xi)¿¿xj¿N(y)f(xj). Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval [1,2n+m-4] and, for trees with k inner vertices, in the interval [1,m+k]. In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree ¿ in the interval [1,n+t(n-t)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n-4k]. In particular, all trees with n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.Proinsulin protects against age-related cognitive loss through anti-inflammatory convergent pathways
http://hdl.handle.net/2117/112127
Proinsulin protects against age-related cognitive loss through anti-inflammatory convergent pathways
Corpas, Ruben; Hernández Pinto, Alberto M.; Porquet, David; Hernández Sánchez, Catalina; Bosch, Fatima; Ortega Aznar, Arantxa; Comellas Padró, Francesc de Paula; de la Rosa, Enrique J.; Sanfeliu Pujol, Coral
Brain inflammaging is increasingly considered as contributing to age-related cognitive loss and neurodegeneration. Despite intensive research in multiple models, no clinically effective pharmacological treatment has been found yet. Here, in the mouse model of brain senescence SAMP8, we tested the effects of proinsulin, a promising neuroprotective agent that was previously proven to be effective in mouse models of retinal neurodegeneration. Proinsulin is the precursor of the hormone insulin but also upholds developmental physiological effects, particularly as a survival factor for neural cells. Adeno-associated viral vectors of serotype 1 bearing the human proinsulin gene were administered intramuscularly to obtain a sustained release of proinsulin into the blood stream, which was able to reach the target area of the hippocampus. SAMP8 mice and the control strain SAMR1 were treated at 1 month of age. At 6 months, behavioral testing exhibited cognitive loss in SAMP8 mice treated with the null vector. Remarkably, the cognitive performance achieved in spatial and recognition tasks by SAMP8 mice treated with proinsulin was similar to that of SAMR1 mice. In the hippocampus, proinsulin induced the activation of neuroprotective pathways and the downstream signaling cascade, leading to the decrease of neuroinflammatory markers. Furthermore, the decrease of astrocyte reactivity was a central effect, as demonstrated in the connectome network of changes induced by proinsulin. Therefore, the neuroprotective effects of human proinsulin unveil a new pharmacological potential therapy in the fight against cognitive loss in the elderly.
2017-12-15T10:16:44ZCorpas, RubenHernández Pinto, Alberto M.Porquet, DavidHernández Sánchez, CatalinaBosch, FatimaOrtega Aznar, ArantxaComellas Padró, Francesc de Paulade la Rosa, Enrique J.Sanfeliu Pujol, CoralBrain inflammaging is increasingly considered as contributing to age-related cognitive loss and neurodegeneration. Despite intensive research in multiple models, no clinically effective pharmacological treatment has been found yet. Here, in the mouse model of brain senescence SAMP8, we tested the effects of proinsulin, a promising neuroprotective agent that was previously proven to be effective in mouse models of retinal neurodegeneration. Proinsulin is the precursor of the hormone insulin but also upholds developmental physiological effects, particularly as a survival factor for neural cells. Adeno-associated viral vectors of serotype 1 bearing the human proinsulin gene were administered intramuscularly to obtain a sustained release of proinsulin into the blood stream, which was able to reach the target area of the hippocampus. SAMP8 mice and the control strain SAMR1 were treated at 1 month of age. At 6 months, behavioral testing exhibited cognitive loss in SAMP8 mice treated with the null vector. Remarkably, the cognitive performance achieved in spatial and recognition tasks by SAMP8 mice treated with proinsulin was similar to that of SAMR1 mice. In the hippocampus, proinsulin induced the activation of neuroprotective pathways and the downstream signaling cascade, leading to the decrease of neuroinflammatory markers. Furthermore, the decrease of astrocyte reactivity was a central effect, as demonstrated in the connectome network of changes induced by proinsulin. Therefore, the neuroprotective effects of human proinsulin unveil a new pharmacological potential therapy in the fight against cognitive loss in the elderly.