Covering times of random walks on bounded degree trees and other graphs

1989 ◽  
Vol 2 (1) ◽  
pp. 147-157 ◽  
Author(s):  
David Zuckerman
Keyword(s):  
Random Walks ◽  
Bounded Degree ◽  
Bounded Degree Trees ◽  
Covering Times

scholarly journals The power of two choices for random walks

2021 ◽  
pp. 1-28
Author(s):  
Agelos Georgakopoulos ◽  
John Haslegrave ◽  
Thomas Sauerwald ◽  
John Sylvester
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Graph ◽  
Optimal Strategy ◽  
Cover Time ◽  
Bounded Degree ◽  
Graph Classes ◽  
Algorithmic Question ◽  
Cover Times ◽  
Bounded Degree Trees

Abstract 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.

Tight Bounds for Embedding Bounded Degree Trees

2010 ◽  
pp. 95-137 ◽  
Author(s):  
Béla Csaba ◽  
Judit Nagy-György ◽  
Ian Levitt ◽  
Endre Szemerédi
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

scholarly journals Helly EPT graphs on bounded degree trees: Characterization and recognition

2017 ◽  
Vol 340 (12) ◽  
pp. 2798-2806
Author(s):  
L. Alcón ◽  
M. Gutierrez ◽  
M.P. Mazzoleni
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

scholarly journals Ramsey Goodness of Bounded Degree Trees

2018 ◽  
Vol 27 (3) ◽  
pp. 289-309 ◽  
Author(s):  
IGOR BALLA ◽  
ALEXEY POKROVSKIY ◽  
BENNY SUDAKOV
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Colour Class ◽  
Bounded Degree Trees

Given a pair of graphsGandH, the Ramsey numberR(G,H) is the smallestNsuch that every red–blue colouring of the edges of the complete graphKNcontains a red copy ofGor a blue copy ofH. If a graphGis connected, it is well known and easy to show thatR(G,H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number ofHand σ(H) is the size of the smallest colour class in a χ(H)-colouring ofH. A graphGis calledH-goodifR(G,H) = (|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then.In this paper we show that ifn≥ Ω(|H| log4|H|) then everyn-vertex bounded degree treeTisH-good. The dependency betweennand |H| is tight up to log factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved thatn-vertex bounded degree trees areH-good whenn≥ Ω(|H|4).

scholarly journals Universal Graphs for Bounded-Degree Trees and Planar Graphs

1989 ◽  
Vol 2 (2) ◽  
pp. 145-155 ◽  
Author(s):  
Sandeep N. Bhatt ◽  
F. R. K. Chung ◽  
F. T. Leighton ◽  
Arnold L. Rosenberg
Keyword(s):  
Planar Graphs ◽  
Bounded Degree ◽  
Universal Graphs ◽  
Bounded Degree Trees

scholarly journals Optimal packings of bounded degree trees

2019 ◽  
Vol 21 (12) ◽  
pp. 3573-3647 ◽  
Author(s):  
Felix Joos ◽  
Jaehoon Kim ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

scholarly journals Recognizing vertex intersection graphs of paths on bounded degree trees

2014 ◽  
Vol 162 ◽  
pp. 70-77 ◽  
Author(s):  
L. Alcón ◽  
M. Gutierrez ◽  
M.P. Mazzoleni
Keyword(s):  
Bounded Degree ◽  
Intersection Graphs ◽  
Bounded Degree Trees

scholarly journals Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

2008 ◽  
Vol 45 (02) ◽  
pp. 481-497 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walks ◽  
Large Class ◽  
Geometric Parameter ◽  
Critical Value ◽  
Critical Values ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Bounded Degree ◽  
Branching Random Walks ◽  
Weak Phase

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

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