COMBGRAF  Combinatòria, Teoria de Grafs i Aplicacions
http://hdl.handle.net/2117/3178
Sat, 20 Jan 2018 17:05:03 GMT
20180120T17:05:03Z

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 nonempty 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.
Fri, 12 Jan 2018 13:52:45 GMT
http://hdl.handle.net/2117/112728
20180112T13:52:45Z
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 nonempty 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.
Wed, 20 Dec 2017 12:55:54 GMT
http://hdl.handle.net/2117/112330
20171220T12:55:54Z
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.

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,…,n1. 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 dki=nifor each0=i=k1, 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.
Wed, 20 Dec 2017 12:04:25 GMT
http://hdl.handle.net/2117/112325
20171220T12:04:25Z
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,…,n1. 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 dki=nifor each0=i=k1, 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+m4] 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(nt)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n4k]. 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/s1099801601512]
Wed, 20 Dec 2017 11:35:12 GMT
http://hdl.handle.net/2117/112321
20171220T11:35:12Z
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+m4] 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(nt)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n4k]. 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 agerelated cognitive loss through antiinflammatory convergent pathways
http://hdl.handle.net/2117/112127
Proinsulin protects against agerelated cognitive loss through antiinflammatory 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 agerelated 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. Adenoassociated 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.
Fri, 15 Dec 2017 10:16:44 GMT
http://hdl.handle.net/2117/112127
20171215T10:16:44Z
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 agerelated 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. Adenoassociated 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.

(Di)graph products, labelings and related results
http://hdl.handle.net/2117/111692
(Di)graph products, labelings and related results
López Masip, Susana Clara
Gallian's survey shows that there is a big variety of labelings of graphs. By means of (di)graphs products we can establish strong relations among some of them. Moreover, due to the freedom of one of the factors, we can also obtain enumerative results that provide lower bounds on the number of nonisomorphic labelings of a particular type. In this paper, we will focus in three of the (di)graphs products that have been used in these duties: the ¿hproduct of digraphs, the weak tensor product of graphs and the weak ¿hproduct of graphs.
Mon, 11 Dec 2017 13:09:06 GMT
http://hdl.handle.net/2117/111692
20171211T13:09:06Z
López Masip, Susana Clara
Gallian's survey shows that there is a big variety of labelings of graphs. By means of (di)graphs products we can establish strong relations among some of them. Moreover, due to the freedom of one of the factors, we can also obtain enumerative results that provide lower bounds on the number of nonisomorphic labelings of a particular type. In this paper, we will focus in three of the (di)graphs products that have been used in these duties: the ¿hproduct of digraphs, the weak tensor product of graphs and the weak ¿hproduct of graphs.

On subsets of the normal rational curve
http://hdl.handle.net/2117/111658
On subsets of the normal rational curve
Ball, Simeon Michael; De Beule, Jan
A normal rational curve of the (k1) dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k6 of a normal rational curve cannot be extended to an arc of size q+2 . In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k3) subset of an arc G of size 3k6 . If G projects to a subset of a conic from every (k3) subset of E , then G cannot be extended to an arc of size q+2 . Stated in terms of errorcorrecting codes we prove that a k dimensional linear maximum distance separable code of length 3k6 over a field Fq of odd characteristic, which can be extended to a Reed–Solomon code of length q+1 , cannot be extended to a linear maximum distance separable code of length q+2 .
Mon, 11 Dec 2017 11:10:59 GMT
http://hdl.handle.net/2117/111658
20171211T11:10:59Z
Ball, Simeon Michael
De Beule, Jan
A normal rational curve of the (k1) dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k6 of a normal rational curve cannot be extended to an arc of size q+2 . In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k3) subset of an arc G of size 3k6 . If G projects to a subset of a conic from every (k3) subset of E , then G cannot be extended to an arc of size q+2 . Stated in terms of errorcorrecting codes we prove that a k dimensional linear maximum distance separable code of length 3k6 over a field Fq of odd characteristic, which can be extended to a Reed–Solomon code of length q+1 , cannot be extended to a linear maximum distance separable code of length q+2 .

Counting configurationfree sets in groups
http://hdl.handle.net/2117/111650
Counting configurationfree sets in groups
Rué Perna, Juan José; Serra Albó, Oriol; Vena Cros, Lluís
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in nonabelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.
Mon, 11 Dec 2017 10:18:23 GMT
http://hdl.handle.net/2117/111650
20171211T10:18:23Z
Rué Perna, Juan José
Serra Albó, Oriol
Vena Cros, Lluís
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in nonabelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.

Characterizing pathlike trees from linear configurations
http://hdl.handle.net/2117/111640
Characterizing pathlike trees from linear configurations
López Masip, Susana Clara
Assume that we embed the path Pn as a subgraph of a 2dimensional grid, namely, Pk¿Pl.
Given such an embedding, we consider the ordered set of subpaths L1, L2,...,Lm which are maximal straight segments in the embedding, and such that the end of Li is the beginning of Li+1.
Mon, 11 Dec 2017 09:01:44 GMT
http://hdl.handle.net/2117/111640
20171211T09:01:44Z
López Masip, Susana Clara
Assume that we embed the path Pn as a subgraph of a 2dimensional grid, namely, Pk¿Pl.
Given such an embedding, we consider the ordered set of subpaths L1, L2,...,Lm which are maximal straight segments in the embedding, and such that the end of Li is the beginning of Li+1.

Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree
http://hdl.handle.net/2117/111531
Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree
Cano Vila, María del Pilar; Perarnau Llobet, Guillem; Serra Albó, Oriol
We study the existence of rainbow perfect matching and rainbow Hamiltonian cycles in edge–colored graphs where every color appears a bounded number of times. We derive asymptotically tight bounds on the minimum degree of the host graph for the existence of such rainbow spanning structures. The proof uses a probabilisitic argument combined with switching techniques.
Mon, 04 Dec 2017 11:51:54 GMT
http://hdl.handle.net/2117/111531
20171204T11:51:54Z
Cano Vila, María del Pilar
Perarnau Llobet, Guillem
Serra Albó, Oriol
We study the existence of rainbow perfect matching and rainbow Hamiltonian cycles in edge–colored graphs where every color appears a bounded number of times. We derive asymptotically tight bounds on the minimum degree of the host graph for the existence of such rainbow spanning structures. The proof uses a probabilisitic argument combined with switching techniques.