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 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 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

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 On Extremal Graphs With No Long Paths

10.37236/1244 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Asad Ali Ali ◽  
William Staton
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Connected Graphs ◽  
Long Paths

Connected graphs with minimum degree $\delta$ and at least $2\delta + 1$ vertices have paths with at least $2\delta + 1$ vertices. We provide a characterization of all such graphs which have no longer paths.

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 $n-1$.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

scholarly journals Matchings in hypergraphs of large minimum degree

2006 ◽  
Vol 51 (4) ◽  
pp. 269-280 ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Minimum Degree

The Performance of Secondary Nasal Alar Base Revision for Unilateral Cleft Lip by Single YV-Plasty (the Importance of Overcorrection During Surgery)

2021 ◽  
pp. 105566562110106
Author(s):  
Yoshitaka Matsuura ◽  
Hideaki Kishimoto
Keyword(s):  
Cleft Lip ◽  
Minimum Degree ◽  
Nasal Deformity ◽  
Simple Method ◽  
Primary Surgery ◽  
Base Revision ◽  
Alar Base ◽  
Anterior Nasal Spine ◽  
Nasal Floor ◽  
Case Basis

Although primary surgery for cleft lip has improved over time, the degree of secondary cleft or nasal deformity reportedly varies from a minimum degree to a remarkable degree. Patients with cleft often worry about residual nose deformity, such as a displaced columella, a broad nasal floor, and a deviation of the alar base on the cleft side. Some of the factors that occur in association with secondary cleft or nasal deformity include a deviation of the anterior nasal spine, a deflected septum, a deficiency of the orbicularis muscle, and a lack of bone underlying the nose. Secondary cleft and nasal deformity can result from incomplete muscle repair at the primary cleft operation. Therefore, surgeons should manage patients individually and deal with various deformities by performing appropriate surgery on a case-by-case basis. In this report, we applied the simple method of single VY-plasty on the nasal floor to a patient with unilateral cleft to revise the alar base on the cleft side. We adopted this approach to achieve overcorrection on the cleft side during surgery, which helped maintain the appropriate position of the alar base and ultimately balanced the nose foramen at 13 months after the operation. It was also possible to complement the height of the nasal floor without a bone graft. We believe that this approach will prove useful for managing cases with a broad and low nasal floor, thereby enabling the reconstruction of a well-balanced nose.

A constructive characterization of contraction critical 8-connected graphs with minimum degree 9

2019 ◽  
Vol 342 (11) ◽  
pp. 3047-3056
Author(s):  
Chengfu Qin ◽  
Weihua He ◽  
Kiyoshi Ando
Keyword(s):  
Minimum Degree ◽  
Connected Graphs
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