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A Classification of Cubic Bicirculants
, 2004
"... The wellknown Petersen G(5, 2) admits a semiregular automorphism α acting on the vertex set with two orbits of equal size. This makes it a bicirculant. It is shown that trivalent bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to ..."
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Cited by 3 (0 self)
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The wellknown Petersen G(5, 2) admits a semiregular automorphism α acting on the vertex set with two orbits of equal size. This makes it a bicirculant. It is shown that trivalent bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conjecture are mentioned.
The Gray Graph Revisited
"... Certain graphtheoretic properties and alternative definitions of ..."
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Cited by 3 (3 self)
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Certain graphtheoretic properties and alternative definitions of
Semisymmetric graphs from polytopes
 J. Combin. Theory Ser. A
, 2007
"... Every finite, selfdual, regular (or chiral) 4polytope of type {3, q,3} has a trivalent 3transitive (or 2transitive) medial layer graph. Here, by dropping selfduality, we obtain a construction for semisymmetric trivalent graphs (which are edge but not vertextransitive). In particular, the Gray ..."
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Cited by 2 (0 self)
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Every finite, selfdual, regular (or chiral) 4polytope of type {3, q,3} has a trivalent 3transitive (or 2transitive) medial layer graph. Here, by dropping selfduality, we obtain a construction for semisymmetric trivalent graphs (which are edge but not vertextransitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes.
The generalized Balaban configurations
, 2001
"... Symmetry properties of the three 10cages on 70 vertices are investigated. Being bipartite, these graphs are Levi graphs of trianglefree and quadranglefree (353 ) configurations. For each of these graphs a Hamilton cycle is given via the associated LCF notation. Furthermore, the automorphism group ..."
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Symmetry properties of the three 10cages on 70 vertices are investigated. Being bipartite, these graphs are Levi graphs of trianglefree and quadranglefree (353 ) configurations. For each of these graphs a Hamilton cycle is given via the associated LCF notation. Furthermore, the automorphism groups of respective orders 80, 120, and 24 are computed. A special emphasis is given to the Balaban 10cage, the first known example of a 10cage [1], and the corresponding Balaban configuration. It is shown that the latter is linear, that is, it can be realized as a geometric configuration of points and lines in the Euclidean plane. Finally, based on the Balaban configuration, an infinite series of linear trianglefree and quadranglefree ((7n)3 ) configurations is produced for each odd integer n 5.
The Remarkable Generalized Petersen Graph G(8, 3)
, 1997
"... Some properties of G(8, 3) are presented showing its uniqueness among generalized Petersen graphs. ..."
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Some properties of G(8, 3) are presented showing its uniqueness among generalized Petersen graphs.
Reductions of (v3) configurations
, 2008
"... Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v3) configurations. In 1887 V. Martinetti described a simple reduction method which enables one to reduce each combinatorial (v3) configuration to one from the infinite set of socalled irreducible configurations. Th ..."
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Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v3) configurations. In 1887 V. Martinetti described a simple reduction method which enables one to reduce each combinatorial (v3) configuration to one from the infinite set of socalled irreducible configurations. The aim of this paper is to show that a slightly extended set of reductions enables one to reduce each combinatorial (v3) configuration either to the Fano configuration or to the Pappus configuration.
A CLASSIFICATION OF CUBIC BICIRCULANTS
, 2004
"... The wellknown Petersen G(5, 2) admits a semiregular automorphism α acting on the vertex set with two orbits of equal size. This makes it a bicirculant. It is shown that trivalent bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to ..."
Abstract
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The wellknown Petersen G(5, 2) admits a semiregular automorphism α acting on the vertex set with two orbits of equal size. This makes it a bicirculant. It is shown that trivalent bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conjecture are mentioned. 1 Introduction and Classification The object of this study are trivalent (cubic) graphs admitting an action of a cyclic group having two equally sized vertex orbits. Such graphs are called bicirculants. The automorphism α that generates the corresponding cyclic group is said to be semiregular. Let G be a bicirculant and let α be the corresponding