Cubic inflation, mirror graphs, regular maps, and partial cubes

Brešar, Boštjan; Klavžar, Sandi; Lipovec, Alenka; Mohar, Bojan

doi:10.1016/j.ejc.2003.09.004

Cited by 13 publications

(23 citation statements)

References 21 publications

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“…The results can be seen as a generalization of results from [3], since all mirror graphs are vertex-transitive partial cubes, and in the cubic case results from [19], since every distance-regular partial cube is vertex-transitive.…”

Section: Introductionmentioning

confidence: 97%

“…It is a well-known result from [15] that hypercubes are the only regular -and thus the only vertex-transitive -median graphs (for infinite vertex-transitive median graphs check [13]). Due to this result, an extensive study of regular partial cubes has been performed [2,3,5,6,11,12]. It especially focuses on the cubic case, since the variety of these graphs is far richer than in the case of median graphs.…”

Section: Introductionmentioning

confidence: 99%

“…From the point of view of vertex-transitive partial cubes, the most interesting one is the study [3], where a new family of graphs called mirror graphs was introduced. Moreover, it was proved that mirror graphs are a subfamily of vertex-transitive partial cubes, and all mirror graphs that are obtained by cubic inflation (thus are cubic graphs) were classified.…”

Section: Introductionmentioning

confidence: 99%

“…It especially focuses on the cubic case, since the variety of these graphs is far richer than in the case of median graphs. Connections with other geometric structures are established, for example with platonic surfaces [3] and simplicial arrangements [6].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classification of Vertex‐Transitive Cubic Partial Cubes

Marc

2017

Journal of Graph Theory

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Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: K 2 C 2n , for some n ≥ 2, the generalized Petersen graph G(10, 3), the cubic permutahedron, the truncated cuboctahedron, or the truncated icosidodecahedron. This classification is a generalization of results of Brešar et al. from 2004 on cubic mirror graphs, it includes all cubic, distance-regular partial cubes (Weichsel, 1992), and presents a contribution to the classification of all cubic partial cubes.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of Vertex‐Transitive Cubic Partial Cubes

Marc

2017

Journal of Graph Theory

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“…For example, [2] address the 3-dimensional case. Also, in [21], an embedding of finitely generated Coxeter group into cube complexes is given.…”

Section: Relation With Coxeter Groupsmentioning

confidence: 99%

Hypercube embedding of Wythoffians

2008

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The Wythoff construction takes a d-dimensional polytope P , a subset S of {0, . . . , d} and returns another d-dimensional polytope P (S). If P is a regular polytope, then P (S) is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4.We want to determine, which of those Wythoffians P (S) with regular P have their skeleton or dual skeleton isometrically embeddable into the hypercubes H m and half-cubes 1 2 H m . We find six infinite series, which, we conjecture, cover all cases for dimension d > 5 and some sporadic cases in dimension 3 and 4 (see Tables 1 and 2).Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable P (S) are addressed throughout the text.

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Hypercube embeddings and Cayley graphs generated by transpositions

Xie

Feng

2022

Math Methods in App Sciences

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A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs—Cayley graphs generated by transpositions—and show that a Cayley graph normalΓ generated by transpositions is a partial cube if and only if normalΓ is a bubble sort graph. This result enhances a result of Alahmadi et al. in 2016: BSn is a partial cube. As a corollary, we give the analytical expressions of the Wiener indices of bubble sort graphs.

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Cubic inflation, mirror graphs, regular maps, and partial cubes

Cited by 13 publications

References 21 publications

Classification of Vertex‐Transitive Cubic Partial Cubes

Classification of Vertex‐Transitive Cubic Partial Cubes

Hypercube embedding of Wythoffians

Hypercube embeddings and Cayley graphs generated by transpositions

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