Rainbow perfect matchings in r-partite graph structures

A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft

Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree

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.Postprint (updated version

On the relation between graph distance and Euclidean distance in random geometric graphs

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).Peer ReviewedPostprint (author's final draft

Matchings in random biregular bipartite graphs

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdös and Rényi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.Peer ReviewedPostprint (published version

Fast recoloring of sparse graphs

This is a post-peer-review, pre-copyedit version of an article published in European Journal of Combinatorics. The final authenticated version is available online at: https://doi.org/10.1016/j.ejc.2015.08.001In this paper, we show that for every graph of maximum average degree bounded away from d and any two (d + 1)-colorings of it, one can transform one coloring into the other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any 8-coloring of a planar graph into any other 8-coloring with a polynomial number of recolorings. These results give some evidence on a conjecture of Cereceda et al [8] which asserts that any (d + 2) coloring of a d-degenerate graph can be transformed into any other one using a polynomial number of recolorings. We also show that any (2d + 2)-coloring of a d-degenerate graph can be transformed into any other one with a linear number of recolorings.Postprint (author's final draft

Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft

Short synchronizing words for random automata

We prove that a uniformly random automaton with n states on a 2-letter alphabet has a synchronizing word of length with high probability (w.h.p.). That is to say, w.h.p. there exists a word ¿ of such length, and a state v0, such that ¿ sends all states to v0. This confirms a conjecture of Kisielewicz, Kowalski, Szykula [KKS13] based on numerical simulations, up to a log factor - the previous best partial result towards the conjecture was the quasilinear bound O(n log3 n) due to Nicaud [Nic19]. Moreover, the synchronizing word ¿ we obtain has small entropy, in the sense that it can be encoded with only O(log(n)) bits w.h.p. Our proof introduces the concept of ¿-trees, for a word ¿, that is, automata in which the ¿-transitions induce a (loop-rooted) tree. We prove a strong structure result that says that, w.h.p., a random automaton on n states is a ¿-tree for some word ¿ of length at most (1 + e) log2(n), for any e > 0. The existence of the (random) word ¿ is proved by the probabilistic method. This structure result is key to proving that a short synchronizing word exists.Postprint (author's final draft

El rol de la depresión en el déficit cognitivo del paciente con síndrome de fatiga crónica

Fundamento y objetivo Analizar el rol de la depresión en el déficit cognitivo del paciente con síndrome de fatiga crónica (SFC). Pacientes y método Un total de 57 mujeres con diagnóstico de SFC fueron evaluadas mediante tests neuropsicológicos que incluían medidas de atención (CalCap, Control Mental del WMS-III, PASAT, dígitos directos e inversos del WAIS-III y symbol digit modalities test [SDMT]) funciones ejecutivas (test Stroop, Trail Making Test [TMT A y B], FAS y Torre de Londres), memoria (Test de Aprendizaje Auditivo-Verbal [TAAVL] y Test de la Figura Compleja de Rey [FCR]) y velocidad psicomotora (Grooved Pegboard). Las puntuaciones directas fueron ajustadas de acuerdo a datos normativos y transformadas a puntuaciones típicas. La muestra fue dividida en dos grupos en función de la presencia o no de depresión, evaluada mediante entrevista clínica y la administración de la Escala Hospitalaria de Ansiedad y Depresión (HAD). Las puntuaciones de los test neuropsicológicos fueron comparadas entre ambos grupos de pacientes. Resultados Los pacientes con SFC presentaron déficit cognitivo en funciones atencionales y ejecutivas, independientemente de la presencia de depresión. No se observaron diferencias significativas en funciones cognitivas entre los dos grupos de pacientes. Conclusiones Estos datos sugieren que el déficit cognitivo que presentan los pacientes con SFC no es secundario a la depresión. Se debería tener en cuenta este resultado en la implementación de un programa terapéutico en estos enfermos

Largest component of subcritical random graphs with given degree sequence

We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.M.C. was supported by the FPI grant PRE2018-083621 associated to the project MTM2017-82166-P from the Spanish Ministerio de Economía y Competitividad. G.P was supported by the Spanish Agencia Estatal de Investigación under projects MTM2017-82166-P and PID2020-113082GB-I00, and by the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).Postprint (author's final draft

Weak components of the directed configuration model

The version of record is available online at: http://dx.doi.org/10.1007/978-3-030-83823-2_109We study the threshold for the existence of a linear order weakly connected component in the directed configuration model, confirming analytic but non-rigorous results recently obtained by Kryven [8]. We also establish convergence in probability of the fraction of vertices and edges that are contained in the largest component. As a consequence of our results, we obtain that the “separation” between the thresholds for the existence a giant weakly and strongly connected component is in some sense independent from the in-/out-degree correlation. We formalise this idea using bond percolation.G. Perarnau—Supported by the Spanish Ministerio de Economía y Competitividad project MTM2017-82166-P and the MSCA-RISE-2020-101007705 - ‘RandNet’.Peer ReviewedPostprint (author's final draft