A Characterization of Astral (n 4 ) Configurations

Berman, Leah Wrenn

doi:10.1007/s00454-001-0041-z

Cited by 20 publications

(37 citation statements)

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“…If the edges in Figure 7 are extended into lines, objects resemble astral configurations of Berman [2] and Grünbaum [12] or polycyclic configurations of Boben and Pisanski [3]. Exploring these intriguing relationships between unit-distance representations of graphs and geometric configurations may be a challenging research project.…”

Section: Resultsmentioning

confidence: 99%

All Generalized Petersen Graphs Are Unit-Distance Graphs

Žitnik¹,

Horvat²,

Pisanski³

2012

Journal of the Korean Mathematical Society

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Abstract. In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n, j, k) admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every I-graph I(n, j, k) has an isomorphic I-graph that admits a unit-distance representation in the Euclidean plane with a n-fold rotational symmetry, with the exception of the families I(n, j, j) and I(12m, m, 5m), m ≥ 1. We also provide unit-distance representations for these graphs.

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Section: Resultsmentioning

confidence: 99%

All Generalized Petersen Graphs Are Unit-Distance Graphs

Žitnik¹,

Horvat²,

Pisanski³

2012

Journal of the Korean Mathematical Society

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“…And similarly, open problems concerning weakly flag-transitive configurations are special cases of open problems on halfarc-transitive graphs (see [16,96,93]). For further directions concerning configurations see [14,18,33,53,54,56,57,113,114].…”

Section: Problemmentioning

confidence: 99%

Recent trends and future directions in vertex-transitive graphs

Kutnar

Marušič

2008

Ars Math. Contemp.

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A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group.Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs -the Hamilton path/cycle problem and the semiregularity problem -as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs. In doing so we will also try to contemplate possible directions this area of research is likely to take in the near future. * Supported by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285.E-mail addresses: klavdija.kutnar@upr.si (Klavdija Kutnar), dragan.marusic@upr.si (Dragan Marušič) Copyright c 2008 DMFA -založništvo K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 113The article is organized as follows. In Section 2 we discuss the problem, posed by the second author in 1981 (see [79]) who asked if it is true that a vertex-transitive digraph contains a nonidentity automorphism with all orbits of equal length, in short, a semiregular automorphism. Section 3 gives a quick overview of the problem, posed by Lovàsz in 1969 (see [72]), who asked if it is true that every connected vertex-transitive graph contains a Hamilton path, thus motivating a great deal of research into vertex-transitive graphs in the following decades. In Section 4, imprimitivity, one of the most fundamental concepts in the theory of permutation groups, is considered with special emphasis given to a recently developed theory which links the existence of blocks of imprimitivity in vertex-transitive graphs, having an abelian semiregular subgroup of automorphisms, to certain conditions that need to be satisfied in the corresponding quotient graph relative to the orbits of such a subgroup. Finally, in Section 5 we deal with some recent structural results on arc-transitive and half-arc-transitive graphs, as well as their link to some open problems in the theory of configurations.For group-theoretic terms not defined here we refer the reader to [124]. Semiregularit...

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“…Although celestial 4-configurations are probably the most well-understood class of 4-configuration, they are still poorly understood in general. The collection of 2-celestial configurations is completely classified ( [2], with a clearer proof in [14, p. 210-211]), but general k-celestial configurations are not completely classified, and the problem appears to be non-tractable (since it depends on being able to solve certain trigonometric diophantine equations). However, some known families of valid k-celestial configurations, primarily for k = 3, 4, were presented in [1].…”

Section: Celestial 4-configurationsmentioning

confidence: 99%

An infinite class of movable 5-configurations

Berman

Jacksch²,

Hoef³

2016

Ars Math. Contemp.

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A geometric 5-configuration is a collection of points and straight lines, typically in the Euclidean plane, in which every point has 5 lines passing through it and every line has 5 points lying on it; that is, it is an (n 5 ) configuration for some number n of points and lines. Using reduced Levi graphs and two elementary geometric lemmas, we develop a construction that produces infinitely many new 5-configurations which are movable; in particular, we produce infinitely many 5-configurations with one continuous degree of freedom, and we produce 5-configurations with k − 2 continuous degrees of freedom for all odd k > 2.

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A Characterization of Astral (n 4 ) Configurations

Abstract: A conjecture of Branko Grünbaum concerning what astral (n 4 ) configurations exist is shown to be true.

Cited by 20 publications

References 1 publication

All Generalized Petersen Graphs Are Unit-Distance Graphs

All Generalized Petersen Graphs Are Unit-Distance Graphs

Recent trends and future directions in vertex-transitive graphs

An infinite class of movable 5-configurations

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