scholarly journals Radius, girth and minimum degree

2022 ◽  
Author(s):  
Vojtĕch Dvorák ◽  
Peter Hintum ◽  
Amy Shaw ◽  
Marius Tiba
Keyword(s):  
10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster

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.


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

2021 ◽  
pp. 105566562110106
Author(s):  
Yoshitaka Matsuura ◽  
Hideaki Kishimoto

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.


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

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

2012 ◽  
Vol 21 (3) ◽  
pp. 457-482 ◽  
Author(s):  
ALLAN LO

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).


2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN

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].


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

Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni

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.


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