Rainbow Connection in Graphs with Minimum Degree Three

Author(s):  
Ingo Schiermeyer
2013 ◽  
Vol 43 (1) ◽  
pp. 7-14 ◽  
Author(s):  
XueLiang LI ◽  
JiuYing DONG

2013 ◽  
Vol 161 (12) ◽  
pp. 1784-1787 ◽  
Author(s):  
Ingo Schiermeyer

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Xiaolong Huang ◽  
Hengzhe Li ◽  
Xueliang Li ◽  
Yuefang Sun

Graph Theory International audience The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.


10.37236/781 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Yair Caro ◽  
Arie Lev ◽  
Yehuda Roditty ◽  
Zsolt Tuza ◽  
Raphael Yuster

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we prove several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee $rc(G)=2$. Among our results we prove that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc(G) < 5n/6$, and if the minimum degree is $\delta$ then $rc(G) \le {\ln \delta\over\delta}n(1+o_\delta(1))$. We also determine the threshold function for a random graph to have $rc(G)=2$ and make several conjectures concerning the computational complexity of rainbow connection.


2016 ◽  
Vol 339 (2) ◽  
pp. 1058-1068 ◽  
Author(s):  
Přemysl Holub ◽  
Zdeněk Ryjáček ◽  
Ingo Schiermeyer ◽  
Petr Vrána

2015 ◽  
Vol 338 (3) ◽  
pp. 1-8 ◽  
Author(s):  
Přemysl Holub ◽  
Zdeněk Ryjáček ◽  
Ingo Schiermeyer

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.


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