scholarly journals From the Coxeter Graph to the Klein Graph

2011 ◽  
Vol 70 (1) ◽  
pp. 1-9
Author(s):  
Italo J. Dejter
Keyword(s):  
Coxeter Graph

Flip-Flops in Hypohamiltonian Graphs

1973 ◽  
Vol 16 (1) ◽  
pp. 33-41 ◽  
Author(s):  
V. Chvátal
Keyword(s):  
Infinite Sequence ◽  
Systematic Search ◽  
Petersen Graph ◽  
Hamiltonian Graph ◽  
Coxeter Graph

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

The Genus of the Coxeter Graph

1995 ◽  
Vol 38 (4) ◽  
pp. 462-464
Author(s):  
Jane M. O. Mitchell
Keyword(s):  
Orientable Surface ◽  
Coxeter Graph

AbstractIn [1], Biggs stated that the Coxeter graph can be embedded in an orientable surface of genus 3. The purpose of this note is to point out that Biggs' embedding is in fact into a non-orientable surface. Further, it is shown that the orientable genus is 3 and the non-orientable genus is 6.

BOUNDARY OF CAT(-1) COXETER GROUPS

1999 ◽  
Vol 09 (02) ◽  
pp. 169-178 ◽  
Author(s):  
N. BENAKLI
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Existence Problem ◽  
Joint Work ◽  
Coxeter Graph ◽  
Combinatorial Properties ◽  
Jsj Decomposition ◽  
Complete Bipartite Graphs ◽  
Dimension 2 ◽  
Virtual Cohomological Dimension

In this paper, we study the topological properties of the hyperbolic boundaries of CAT(-1) Coxeter groups of virtual cohomological dimension 2. We will show how these properties are related to combinatorial properties of the associated Coxeter graph. More precisely, we investigate the connectedness, the local connectedness and the existence problem of local cut points. In the appendix, in a joint work with Z. Sela, we will construct the JSJ decomposition of the Coxeter groups for which the corresponding Coxeter graphs are complete bipartite graphs.

On the crystallography of infinite Coxeter groups

1977 ◽  
Vol 82 (1) ◽  
pp. 13-24 ◽  
Author(s):  
George Maxwell
Keyword(s):  
Affine Space ◽  
Coxeter Groups ◽  
Root Systems ◽  
Space Groups ◽  
Coxeter Graph ◽  
Finite Dimensional ◽  
Multipartite Graphs ◽  
Real Affine ◽  
Point Groups ◽  
Complete Multipartite

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

scholarly journals On Generalizations of the Petersen Graph and the Coxeter Graph

10.37236/3759 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Marko Orel
Keyword(s):  
Petersen Graph ◽  
Coxeter Graph

In this note we consider two related infinite families of graphs, which generalize the Petersen and the Coxeter graph. The main result proves that these graphs are cores. It is determined which of these graphs are vertex/edge/arc-transitive or distance-regular. Girths and odd girths are computed. A problem on hamiltonicity is posed.A Corrigendum for this paper was added on August 19, 2017.

A GEOMETRIC AND ALGEBRAIC DESCRIPTION OF ANNULAR BRAID GROUPS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 85-97 ◽  
Author(s):  
RICHARD P. KENT ◽  
DAVID PEIFER
Keyword(s):  
Finite Type ◽  
Cyclic Group ◽  
Semidirect Product ◽  
Braid Group ◽  
Braid Groups ◽  
Artin Group ◽  
Infinite Cyclic Group ◽  
Coxeter Graph ◽  
Infinite Type ◽  
Algebraic Description

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn - 1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph Ãn - 1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear.

scholarly journals Note on the residual finiteness of Artin groups

2018 ◽  
Vol 21 (3) ◽  
pp. 531-537 ◽  
Author(s):  
Rubén Blasco-García ◽  
Arye Juhász ◽  
Luis Paris
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Coxeter Graph ◽  
Admissible Partition ◽  
Group A

Abstract Let A be an Artin group. A partition {\mathcal{P}} of the set of standard generators of A is called admissible if, for all {X,Y\in\mathcal{P}} , {X\neq Y} , there is at most one pair {(s,t)\in X\times Y} which has a relation. An admissible partition {\mathcal{P}} determines a quotient Coxeter graph {\Gamma/\mathcal{P}} . We prove that, if {\Gamma/\mathcal{P}} is either a forest or an even triangle free Coxeter graph and {A_{X}} is residually finite for all {X\in\mathcal{P}} , then A is residually finite.

ORDERINGS ON ARTIN–TITS GROUPS

2008 ◽  
Vol 18 (06) ◽  
pp. 1035-1066
Author(s):  
HERVE SIBERT
Keyword(s):  
Partial Order ◽  
Partial Ordering ◽  
Necessary Condition ◽  
Coxeter Graph ◽  
Type D ◽  
Tits Group

We prove that a construction similar to that described by Dehornoy in the case of braids is possible for every Artin–Tits group, yielding a partial ordering. A necessary condition for this partial order to be linear is that the associated Coxeter graph consists only of disjoint lines. So, in particular, type D is dismissed.

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