scholarly journals Conflict-free vertex-connections of graphs

2020 ◽  
Vol 40 (1) ◽  
pp. 51 ◽  
Author(s):  
Stanislav Jendrol' ◽  
Xueliang Li ◽  
Yaping Mao ◽  
Yingying Zhang ◽  
Haixing Zhao ◽  
...  
Keyword(s):  
Free Vertex

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

The 1-Good-Neighbor Conditional Diagnosability of Some Regular Graphs

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741001
Author(s):  
MEI-MEI GU ◽  
RONG-XIA HAO ◽  
AI-MEI YU
Keyword(s):  
Alternating Group ◽  
Regular Graphs ◽  
Connected Graphs ◽  
Free Vertex ◽  
Pmc Model ◽  
Star Networks ◽  
Conditional Diagnosability ◽  
Good Neighbor

The g-good-neighbor conditional diagnosability is the maximum number of faulty vertices a network can guarantee to identify, under the condition that every fault-free vertex has at least g fault-free neighbors. In this paper, we study the 1-good-neighbor conditional diagnosabilities of some general k-regular k-connected graphs G under the PMC model and the MM* model. The main result [Formula: see text] under some conditions is obtained, where l is the maximum number of common neighbors between any two adjacent vertices in G. Moreover, the following results are derived: [Formula: see text] for the hierarchical star networks, [Formula: see text] for the BC networks, [Formula: see text] for the alternating group graphs [Formula: see text].

scholarly journals A Deviation-Based Dynamic Vertex Reordering Technique for 2D Mesh Quality Improvement

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 895 ◽  
Author(s):  
Junhyeok Choi ◽  
Harrim Kim ◽  
Shankar Prasad Sastry ◽  
Jibum Kim
Keyword(s):  
Quality Improvement ◽  
Optimization Problem ◽  
Mesh Optimization ◽  
Mesh Quality ◽  
Free Vertex ◽  
Smooth Optimization ◽  
Downhill Simplex ◽  
Mesh Quality Improvement ◽  
Better Than

We propose a novel deviation-based vertex reordering method for 2D mesh quality improvement. We reorder free vertices based on how likely this is to improve the quality of adjacent elements, based on the gradient of the element quality with respect to the vertex location. Specifically, we prioritize the free vertex with large differences between the best and the worst-quality element around the free vertex. Our method performs better than existing vertex reordering methods since it is based on the theory of non-smooth optimization. The downhill simplex method is employed to solve the mesh optimization problem for improving the worst element quality. Numerical results show that the proposed vertex reordering techniques improve both the worst and average element, compared to those with existing vertex reordering techniques.

scholarly journals Combinatorics of free vertex algebras

2002 ◽  
Vol 255 (2) ◽  
pp. 297-323 ◽  
Author(s):  
Michael Roitman
Keyword(s):  
Vertex Algebras ◽  
Free Vertex

On square-free vertex colorings of graphs

2007 ◽  
Vol 44 (3) ◽  
pp. 411-422 ◽  
Author(s):  
János Barát ◽  
Péter Varjú
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Path Following ◽  
Fixed Number ◽  
The Other ◽  
Graph Colorings ◽  
Outerplanar Graphs ◽  
Free Vertex ◽  
Other Hand ◽  
Color Sequence

A sequence of symbols a1 , a2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x1 , …, xm , x1 , …, xm . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Hałuszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( ck ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

Maximum value of conflict-free vertex-connection number of graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850059 ◽  
Author(s):  
Zhenzhen Li ◽  
Baoyindureng Wu
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Colored Graph ◽  
Connection Number ◽  
Free Vertex ◽  
Maximum Value

A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

scholarly journals Conflict-free vertex connection number at most 3 and size of graphs

2021 ◽  
Vol 41 (2) ◽  
pp. 617 ◽  
Author(s):  
Trung Duy Doan ◽  
Ingo Schiermeyer
Keyword(s):  
Connection Number ◽  
Free Vertex

scholarly journals On the connectivity of p-diamond-free vertex transitive graphs

2012 ◽  
Vol 160 (7-8) ◽  
pp. 1285-1290
Author(s):  
Yingzhi Tian ◽  
Jixiang Meng ◽  
Zhao Zhang
Keyword(s):  
Free Vertex ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

scholarly journals Vertex-Decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity

10.37236/2387 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Türker Bıyıkoğlu ◽  
Yusuf Civan
Keyword(s):  
Vertex Cover ◽  
Infinite Family ◽  
Matching Number ◽  
Induced Matching ◽  
Free Vertex ◽  
Acyclic Digraph ◽  
Vertex Decomposable ◽  
Decomposable Graphs

We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, and a graph $G$ is called codismantlable if either it is an edgeless graph or it contains a codominated vertex $x$ such that $G-x$ is codismantlable. We show that $(C_4,C_5)$-free vertex-decomposable graphs are codismantlable, and prove that if $G$ is a $(C_4,C_5,C_7)$-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for $G$ are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity $reg(G)$ of such graphs, and show that $reg(G)=im(G)$ whenever $G$ is a $(C_4,C_5)$-free vertex-decomposable graph, where $im(G)$ is the induced matching number of $G$. Furthermore, we prove that $H$ must be a codismantlable graph if $im(H)=reg(H)=m(H)$, where $m(H)$ is the matching number of $H$. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family $H_n$ ($n\geq 4$) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and $reg(H_n)=im(H_n)$ if $n\geq 6$. This answers a recent question of Mahmoudi et al.

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