Long cycles in subgraphs of (pseudo)random directed graphs

Random Struct Algorithms

2013

Self Cite

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p≥ω/k for a function ω=ω(k) that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least (1−o(1))k, and (ii) if p≥(1+o(1))lnk/k, then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015

Section: For Every Subsetmentioning

confidence: 99%

“…The main technique we use in proving our theorems is a technique recently developed in Refs. and , based on the depth first search algorithm. In Section 2, we discuss this technique in detail and also provide some probabilistic tools that we will need later.…”

Section: Introductionmentioning

confidence: 99%

Long paths and cycles in random subgraphs of graphs with large minimum degree

Krivelevich

Lee

Random Struct Algorithms

2013

Self Cite

“…The following lemma is due to Ben‐Eliezer, Krivelevich, and Sudakov and is a useful application of the Depth‐first search (DFS) algorithm.Lemma Given an oriented graph G, there is a directed path P such that vertices of G – P can be partitioned into two disjoint sets U and W such that

| U | = | W |

and all the edges between U and W are oriented from W to U.Proof We start with

U = \emptyset, W = V (G)

and

P = \emptyset .

We repeat the following procedure, throughout of which P is a path,

| W | \geq | U |

and all edges between U and W are oriented from W to U . If P is empty we take any vertex from W to be the new path P and remove the vertex from W .…”

Section: Preliminariesmentioning

confidence: 99%

Monochromatic paths in random tournaments

Bucić

Letzter

Random Struct Algorithms

2018

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“…The following lemma, based on the Depth-first-search (DFS) algorithm, is an easy generalisation, for trees, of the version for paths introduced by Ben-Eliezer, Krivelevich and Sudakov [3,4]; it will be very useful in several parts of this section.…”

Section: Preliminariesmentioning

confidence: 99%

Directed Ramsey number for trees

Bucić

Letzter

Electronic Notes in Discrete Mathematics

2017

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In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by − → r (H, k), which is the least n for which every k-edgecoloured tournament on n vertices contains a monochromatic copy of H. We prove that − → r (T, k) ≤ c k |T | k for any oriented tree T . This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor.We also consider the k-colour directed Ramsey number ← → r (H, k) of H, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order n. Here we show that ← → r (T, k) ≤ c k |T | k−1 for any oriented tree T , which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the 2-colour directed Ramsey number of directed paths.