scholarly journals Simple graphs of order 12 and minimum degree 6 contain K6 minors

2020 ◽  
Vol 13 (5) ◽  
pp. 829-843
Author(s):  
Ryan Odeneal ◽  
Andrei Pavelescu
Keyword(s):  
Minimum Degree ◽  
Simple Graphs

scholarly journals A Generalization of Stiebitz-Type Results on Graph Decomposition

10.37236/9757 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Qinghou Zeng ◽  
Chunlei Zu
Keyword(s):  
Minimum Degree ◽  
Graph Decomposition ◽  
Simple Graphs

In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $\mu_G(v)=\max\{\mu_G(u,v):u\in V(G)\setminus\{v\}\}$, where $\mu_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2\mu_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(x)\geq a(x)$ for each $x\in X$ and $d_Y(y)\geq b(y)$ for each $y\in Y$. This extends the related results due to Diwan, Liu–Xu and  Ma–Yang on simple graphs to the multigraph setting.

scholarly journals A rainbow version of Mantel's Theorem

2020 ◽  
Author(s):  
Robert Šámal ◽  
Amanda Montejano ◽  
Sebastián González Hermosillo de la Maza ◽  
Matt DeVos ◽  
Ron Aharoni
Keyword(s):  
Graph Theory ◽  
Minimum Degree ◽  
Computer Assisted ◽  
Simple Graphs ◽  
Computer Assisted Proof ◽  
Color Class ◽  
Vertex Graph ◽  
Degree Conditions

Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, which can be viewed as a union of simple graphs, each corresponding to an edge-color class. The case of two colors is the same as the original setting: Diwan and Mubayi proved that any two graphs $G_1$ and $G_2$ on the same set of $n$ vertices, each containing more than $\frac{1}{4}n^2$ edges, give rise to a triangle with one edge from $G_1$ and two edges from $G_2$. The situation is however different for three colors. Fix $\tau=\frac{4-\sqrt{7}}{9}$ and split the $n$ vertices into three sets $A$, $B$ and $C$, such that $|B|=|C|=\lfloor\tau n\rfloor$ and $|A|=n-|B|-|C|$. The graph $G_1$ contains all edges inside $A$ and inside $B$, the graph $G_2$ contains all edges inside $A$ and inside $C$, and the graph $G_3$ contains all edges between $A$ and $B\cup C$ and inside $B\cup C$. It is easy to check that there is no triangle with one edge from $G_1$, one from $G_2$ and one from $G_3$; each of the graphs has about $\frac{1+\tau^2}{4}n^2=\frac{26-2\sqrt{7}}{81}n^2\approx 0.25566n^2$ edges. The main result of the article is that this construction is optimal: any three graphs $G_1$, $G_2$ and $G_3$ on the same set of $n$ vertices, each containing more than $\frac{1+\tau^2}{4}n^2$ edges, give rise to a triangle with one edge from each of the graphs $G_1$, $G_2$ and $G_3$. A computer-assisted proof of the same bound has been found by Culver, Lidický, Pfender and Volec.

Super Graceful Labeling for Some Simple Graphs

2012 ◽  
Vol 2 (1) ◽  
pp. 35 ◽  
Author(s):  
M. A. Perumal ◽  
S. Navaneethakrishnan ◽  
A. Nagaraja ◽  
S. Arockiaraj
Keyword(s):  
Graceful Labeling ◽  
Simple Graphs

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

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

Design Criteria for Safer Manual Lifting by Men and Women

1982 ◽  
Vol 26 (6) ◽  
pp. 503-507
Author(s):  
Dudley G. Letbetter
Keyword(s):  
Muscle Fatigue ◽  
Material Handling ◽  
Human Anatomy ◽  
Design Criteria ◽  
Maximum Weight ◽  
Manual Lifting ◽  
Men And Women ◽  
Simple Graphs ◽  
Cardiovascular Capacity ◽  
Specific Material

Simplified design criteria are provided for two-handed, manual lifting by standing men and women, without selective assignment of personnel to specific material handling tasks. Based on a 1981 NIOSH report, application of these criteria requires no knowledge of human anatomy, anthropometry, biomechanics, psychophysics, muscle fatigue, cardiovascular capacity, or metabolic endurance. A person who can read and use simple graphs can quickly determine the maximum weight of a lifted object. The information needed is the horizontal grasp distance and the initial grasp height and lift distance of the object, plus the frequency and duration of lifting.

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