Probabilistic methods and coloring problems in graphs

Aquest projecte està dedicat a estudiar el k-èssim nombre cromàtic generalitzat que sorgeix de les descomposicions Low Tree--Depth en grafs usant mètodes probabilístics.. Una extensió natural del nombre cromàtic d'un graf és l'estudi de particions de grafs en les que cada i parts indueixen un subgraf amb un cert paràmetre acotat en funció de i, per exemple cada i parts tenen com a molt i-1 arestes. En particular el nombre cromàtic generalitzat és le mínim nombre de parts per tal que cada i parts té 'treedepth' com a molt i. Resultats recents proven que grans classes de grafs tenen paràmetres d'aquest tipus acotats. L'objectiu del projecte és (i) fer servie mètodes probabilístics per donar cotas ajustades d'aquests paràmetres i (ii) estudiar el seu valor per grafs aleatoris

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

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

Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture

A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that if is any bridge-addable class of graphs on n vertices, and is taken uniformly at random from , then is connected with probability at least , when n tends to infinity. This lower bound is asymptotically best possible since it is reached for forests. Our proof uses a “local double counting” strategy that may be of independent interest, and that enables us to compare the size of two sets of combinatorial objects by solving a related multivariate optimization problem. In our case, the optimization problem deals with partition functions of trees relative to a supermultiplicative functional.Postprint (author's final draft

The giant component of the directed configuration model revisited

We prove a law of large numbers for the order and size of the largest strongly connected component in the directed configuration model. Our result extends previous work by Cooper and Frieze (2004).Supported by the Spanish Ministerio de Economía y Competitividad project MTM2017- 82166-P and the MSCA-RISE-2020-101007705 - ’RandNet’. 1517Postprint (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

Percolation on random graphs with a fixed degree sequence

We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterization of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique which is based on the switching method and avoids the use of the classic configuration model on degree sequences that have a limiting distribution. Thus our results hold for sparse degree sequences without the usual restrictions that accompany the configuration model.The research was also supported by the EPSRC, grant no. EP/M009408/1.Postprint (author's final draft

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

Continuous symmetry and shape measures

Postprint (published version