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8686 results in Discrete Mathematics, Information Theory and Coding

Page 48 of 435

  • The power of two choices for random walks

  • Part of
  • Agelos Georgakopoulos, John Haslegrave, Thomas Sauerwald, John Sylvester
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 1 / January 2022
    Published online by Cambridge University Press:
    28 May 2021, pp. 73-100
    • Article
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  • We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

  • Spanning trees in random regular uniform hypergraphs

  • Part of
  • Catherine Greenhill, Mikhail Isaev, Gary Liang
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 1 / January 2022
    Published online by Cambridge University Press:
    26 May 2021, pp. 29-53

  • Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$, restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

Page 48 of 435