10 research outputs found
Semisymmetric graphs from polytopes
AbstractEvery finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs
It is known that the Levi graph of any rank two coset geometry is an
edge-transitive graph, and thus coset geometries can be used to construct many
edge transitive graphs. In this paper, we consider the reverse direction.
Starting from edge- transitive graphs, we construct all associated core-free,
rank two coset geometries. In particular, we focus on 3-valent and 4-valent
graphs, and are able to construct coset geometries arising from these graphs.
We summarize many properties of these coset geometries in a sequence of tables;
in the 4-valent case we restrict to graphs that have relatively small
vertex-stabilizers
Resolution of a conjecture about linking ring structures
An LR-structure is a tetravalent vertex-transitive graph together with a
special type of a decomposition of its edge-set into cycles. LR-structures were
introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings
structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7
(2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this
paper, we use the methods of group amalgams to resolve some problems left open
in the above-mentioned paper
Semiregular Polytopes and Amalgamated C-groups
In the classical setting, a convex polytope is said to be semiregular if its
facets are regular and its symmetry group is transitive on vertices. This paper
studies semiregular abstract polytopes, which have abstract regular facets,
still with combinatorial automorphism group transitive on vertices. We analyze
the structure of the automorphism group, focusing in particular on polytopes
with two kinds of regular facets occurring in an "alternating" fashion. In
particular we use group amalgamations to prove that given two compatible
n-polytopes P and Q, there exists a universal abstract semiregular
(n+1)-polytope which is obtained by "freely" assembling alternate copies of P
and Q. We also employ modular reduction techniques to construct finite
semiregular polytopes from reflection groups over finite fields.Comment: Advances in Mathematics (to appear, 28 pages