Capítols de llibrehttp://hdl.handle.net/2117/3650102024-07-12T23:34:25Z2024-07-12T23:34:25ZTutte uniqueness and Tutte equivalenceBonin, Joseph E.Mier Vinué, Anna dehttp://hdl.handle.net/2117/3830082023-02-14T15:20:19Z2023-02-14T15:17:22ZTutte uniqueness and Tutte equivalence
Bonin, Joseph E.; Mier Vinué, Anna de
This chapter considers graphs or matroids that have the same Tutte polynomial, as well as graphs or matroids that, up to isomorphism, are distinguished from all others by their Tutte polynomial. We call the former Tutte equivalent, and the latter Tutte unique. Tutte invariants (data that the Tutte polynomial contains). Operations that preserve Tutte uniqueness or equivalence. Connections between the graph and matroid Tutte-uniqueness problems. Constructions of large families of Tutte-equivalent graphs and matroids. Tutte-unique graphs and matroids, and characterizations of some of these graphs and matroids by Tutte invariants. Related lines of research.
2023-02-14T15:17:22ZBonin, Joseph E.Mier Vinué, Anna deThis chapter considers graphs or matroids that have the same Tutte polynomial, as well as graphs or matroids that, up to isomorphism, are distinguished from all others by their Tutte polynomial. We call the former Tutte equivalent, and the latter Tutte unique. Tutte invariants (data that the Tutte polynomial contains). Operations that preserve Tutte uniqueness or equivalence. Connections between the graph and matroid Tutte-uniqueness problems. Constructions of large families of Tutte-equivalent graphs and matroids. Tutte-unique graphs and matroids, and characterizations of some of these graphs and matroids by Tutte invariants. Related lines of research.An approximate structure theorem for small sumsetsCampos, MarceloCoulson, Matthew JohnSerra Albó, OriolWötzel, Maximilianhttp://hdl.handle.net/2117/3692732023-08-25T00:27:42Z2022-06-29T09:49:52ZAn approximate structure theorem for small sumsets
Campos, Marcelo; Coulson, Matthew John; Serra Albó, Oriol; Wötzel, Maximilian
Let A and B be randomly chosen s-subsets of the first n integers such that their sumset A+B has size at most Ks. We show that asymptotically almost surely A and B are almost fully contained in arithmetic progressions PA and PB with the same common difference and cardinalities approximately Ks/2. The result holds for s=¿(log3n) and 2=K=o(s/log3n). Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove the result in the special case A=B.
2022-06-29T09:49:52ZCampos, MarceloCoulson, Matthew JohnSerra Albó, OriolWötzel, MaximilianLet A and B be randomly chosen s-subsets of the first n integers such that their sumset A+B has size at most Ks. We show that asymptotically almost surely A and B are almost fully contained in arithmetic progressions PA and PB with the same common difference and cardinalities approximately Ks/2. The result holds for s=¿(log3n) and 2=K=o(s/log3n). Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove the result in the special case A=B.The expected number of perfect matchings in cubic planar graphsRué Perna, Juan JoséRequile, ClementNoy Serrano, Marcoshttp://hdl.handle.net/2117/3650092022-08-07T06:22:49Z2022-03-30T10:04:20ZThe expected number of perfect matchings in cubic planar graphs
Rué Perna, Juan José; Requile, Clement; Noy Serrano, Marcos
A well-known conjecture by Lovász and Plummer from the 1970s asserting that a bridgeless cubic graph has exponentially many perfect matchings was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture for the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically c¿n, where c>0 and ¿~1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations. (Supported by the Ministerio de Economía y Competitividad grant MTM2017-82166-P, and by the Special Research Program F50 Algorithmic and Enumerative Combinatorics of the Austrian Science Fund.).
2022-03-30T10:04:20ZRué Perna, Juan JoséRequile, ClementNoy Serrano, MarcosA well-known conjecture by Lovász and Plummer from the 1970s asserting that a bridgeless cubic graph has exponentially many perfect matchings was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture for the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically c¿n, where c>0 and ¿~1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations. (Supported by the Ministerio de Economía y Competitividad grant MTM2017-82166-P, and by the Special Research Program F50 Algorithmic and Enumerative Combinatorics of the Austrian Science Fund.).Asymptotics of sequencesBender, EdwardRué Perna, Juan Joséhttp://hdl.handle.net/2117/1164382020-07-23T21:35:55Z2018-04-18T11:49:39ZAsymptotics of sequences
Bender, Edward; Rué Perna, Juan José
2018-04-18T11:49:39ZBender, EdwardRué Perna, Juan José