Empacotamento e contagem em digrafos: cenários aleatórios e extremais

Abstract

In this thesis we study two problems dealing with digraphs: a packing problem and a counting problem. We study the problem of packing the maximum number of arborescences in the random digraph D(n, p), where each possible arc is included uniformly at random with probability p = p(n). Let λ(D(n, p)) denote the largest integer λ ≥ 0 such that, for all 0 ≤ ≤ λ, we have −1 i=0 (− i)|{v : d in (v) = i}| ≤. We show that the maximum number of arc-disjoint arborescences in D(n, p) is λ(D(n, p)) asymptotically almost surely. We also give tight estimates for λ(D(n, p)) for every p ∈ [0, 1]. The main tools that we used were expansion properties of random digraphs, the behavior of in-degree of random digraphs and a classic result by Frank relating subpartitions and number of arborescences. For the counting problem, we study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct of this study we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5. We use the theory of flag algebras as a main tool for this problem and combinatorial settings obtained from semidefinite method.

Utilizando álgebras de flags para problemas de combinatória extremal

“…Neste trabalho, apresentamos uma breve introduçãoà teoria deálgebras de flag, desenvolvida por Razborov [Razborov 2007], que tem sido utilizada em diversos problemas de combinatória extremal. Para ilustrar a aplicabilidade dessa teoria, apresentamos um resultado para torneios [Coregliano et al 2015, Parente 2016] obtido utilizando técnicas da teoria deálgebra de flags.…”

“…Para que A σ seja de fato umaálgebra,é preciso que definir também um produto. Para detalhes, veja o artigo[Coregliano et al 2015, Parente 2016]. 2 Uma função φ : A σ → Ré um homomorfismo deálgebra se para todo α ∈ R e f, g ∈ A σ , vale queφ(α • f ) = α • φ(f ), φ(f • g) = φ(f ) • φ(g) e φ(f + g) = φ(f ) + φ(g).…”