scholarly journals Long cycles in random subgraphs of graphs with large minimum degree

2014 ◽  
Vol 45 (4) ◽  
pp. 764-767 ◽  
Author(s):  
Oliver Riordan
Keyword(s):  
Minimum Degree ◽  
Long Cycles ◽  
Random Subgraphs

scholarly journals Long cycles in subgraphs with prescribed minimum degree

1991 ◽  
Vol 97 (1-3) ◽  
pp. 69-81 ◽  
Author(s):  
L. Caccetta ◽  
K. Vijayan
Keyword(s):  
Minimum Degree ◽  
Long Cycles

scholarly journals Long cycles in hypercubes with distant faulty vertices

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
International Audience ◽  
Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

scholarly journals Long Cycles in t-Tough Graphs with t > 1

2019 ◽  
pp. 39-56
Author(s):  
Zhora Nikoghosyan
Keyword(s):  
Minimum Degree ◽  
Petersen Graph ◽  
Long Cycles

It is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graphIt is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graph

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

scholarly journals Large complete minors in random subgraphs

2020 ◽  
pp. 1-12
Author(s):  
Joshua Erde ◽  
Mihyun Kang ◽  
Michael Krivelevich
Keyword(s):  
High Probability ◽  
Minimum Degree ◽  
Tight Bound ◽  
Constant Multiple ◽  
Random Subgraph ◽  
Random Subgraphs

Abstract Let G be a graph of minimum degree at least k and let G p be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that G p contains when p = (1 + ε)/k with ε > 0. We show that with high probability G p contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.

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

2013 ◽  
Vol 46 (2) ◽  
pp. 320-345 ◽  
Author(s):  
Michael Krivelevich ◽  
Choongbum Lee ◽  
Benny Sudakov
Keyword(s):  
Minimum Degree ◽  
Random Subgraphs ◽  
Long Paths

scholarly journals Long cycles in graphs with prescribed toughness and minimum degree

1995 ◽  
Vol 141 (1-3) ◽  
pp. 1-10 ◽  
Author(s):  
Douglas Bauer ◽  
H.J. Broersma ◽  
J. van den Heuvel ◽  
H.J. Veldman
Keyword(s):  
Minimum Degree ◽  
Long Cycles

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Extreme degrees in random subgraphs of regular graphs

1985 ◽  
Vol 97 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Minimum Degree ◽  
Probability Distributions ◽  
Asymptotic Distributions ◽  
Regular Graphs ◽  
Additional Information ◽  
Initial Graph ◽  
Random Subgraph ◽  
Edge Probability ◽  
Maximum And Minimum Degree ◽  
Random Subgraphs

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

scholarly journals Paths and Cycles in Random Subgraphs of Graphs with Large Minimum Degree

10.37236/6761 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Stefan Ehard ◽  
Felix Joos
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Graph Model ◽  
Arbitrary Graph ◽  
Random Graph Model ◽  
Longest Path ◽  
Random Subgraphs

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$.For a constant $c>0$ and $p=\frac{c}{n}$, we show that asymptotically almost surely the length of the longest path in $G_p$ is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function $\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically best-possible. This extends several known results on the length of the longest path/cycle of a random graph in the $G(n,p)$-model to the random graph model $G_p$ where $G$ is a graph of minimum degree at least $n-1$.

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