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 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.

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

scholarly journals Construction of planar triangulations with minimum degree 5

2005 ◽  
Vol 301 (2-3) ◽  
pp. 147-163 ◽  
Author(s):  
G. Brinkmann ◽  
Brendan D. McKay
Keyword(s):  
Minimum Degree

scholarly journals Cliques in Graphs With Bounded Minimum Degree

2012 ◽  
Vol 21 (3) ◽  
pp. 457-482 ◽  
Author(s):  
ALLAN LO
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Minimum Number

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals The minimum degree distance of graphs of given order and size

2008 ◽  
Vol 156 (18) ◽  
pp. 3518-3521 ◽  
Author(s):  
Orest Bucicovschi ◽  
Sebastian M. Cioabă
Keyword(s):  
Minimum Degree ◽  
Degree Distance

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

Scalable mining of maximal quasi-cliques

2020 ◽  
Vol 14 (4) ◽  
pp. 573-585
Author(s):  
Guimu Guo ◽  
Da Yan ◽  
M. Tamer Özsu ◽  
Zhe Jiang ◽  
Jalal Khalil
Keyword(s):  
Graph Mining ◽  
State Of The Art ◽  
Minimum Degree ◽  
The Other ◽  
Dense Subgraph ◽  
Load Imbalance ◽  
Subgraph Mining ◽  
Execution Engine ◽  
Clique Algorithm ◽  
Dense Subgraph Mining

Given a user-specified minimum degree threshold γ , a γ -quasiclique is a subgraph g = (V g , E g ) where each vertex ν ∈ V g connects to at least γ fraction of the other vertices (i.e., ⌈ γ · (| V g |- 1)⌉ vertices) in g. Quasi-clique is one of the most natural definitions for dense structures useful in finding communities in social networks and discovering significant biomolecule structures and pathways. However, mining maximal quasi-cliques is notoriously expensive. In this paper, we design parallel algorithms for mining maximal quasi-cliques on G-thinker, a distributed graph mining framework that decomposes mining into compute-intensive tasks to fully utilize CPU cores. We found that directly using G-thinker results in the straggler problem due to (i) the drastic load imbalance among different tasks and (ii) the difficulty of predicting the task running time. We address these challenges by redesigning G-thinker's execution engine to prioritize long-running tasks for execution, and by utilizing a novel timeout strategy to effectively decompose long-running tasks to improve load balancing. While this system redesign applies to many other expensive dense subgraph mining problems, this paper verifies the idea by adapting the state-of-the-art quasi-clique algorithm, Quick, to our redesigned G-thinker. Extensive experiments verify that our new solution scales well with the number of CPU cores, achieving 201× runtime speedup when mining a graph with 3.77M vertices and 16.5M edges in a 16-node cluster.

Sign in / Sign up

Export Citation Format

Share Document