Abstract
A connected graph G is called \(l_{1}\)-embeddable, if it can be isometrically embedded into the \(l_{1}\)-space. The shifted quadrilateral cylinder graph \(O_{m,n,k}\) is a class of quadrilateral tilings on a cylinder obtained by rolling the grid graph \(P_{m}\square P_{n}\) via shifting k positions. In this article, we determine that all the \(O_{m,n,k}\) are not \(l_{1}\)-embeddable except for \(O_{m,n,0}\) and \(O_{m,3,1}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statements
All data generated or analysed during this study are included in this published article.
References
Assouad, P., Deza, M.: Espaces métriques plongeables dans un hypercube: aspects combinatoires. Ann. Discrete Math. 8, 197–210 (1980)
Bandelt, H.J., Chepoi, V.: Decomposition and \(l_1\)-embedding of weakly median graphs. Eur. J. Combin. 21, 701–714 (2000)
Bandelt H.J., Chepoi V.: Metric graph theory and geometry: a survey. In J. E. Goodman, J. Pach, and R. Pollack, editors, Surveys on Discrete and Computational Geometry: Twenty Years Later, volume 453 of Contemp. Math., pages 49C86. Amer. Math. Soc., Providence, RI, 2008. https://doi.org/10.1090/conm/453/08795.
Chepoi, V.: Basis graphs of even Delta-matroids. J. Comb. Theory Ser. B 97, 175–192 (2007)
Chepoi, V.: Distance-preserving subgraphs of Johnson graphs. Combinatorica 37(6), 1039–1055 (2017)
Chepoi, V., Deza, M., Grishukhin, V.: Clin d’oeil on \(l_1\)-embeddable planar graphs. Discrete Appl. Math. 80, 3–19 (1997)
Deza, M., Laurent, M.: \(l_1\)-rigid graphs. J. Algebra Comb. 3, 153–175 (1994)
Deza, M., Shpectorov, S.: Recognition of the \(l_1\)-graphs with complexity \(O\)(\(nm\)), or football in a hypercube. Eur. J. Combin. 17, 279–289 (1996)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer-Verlag, Berlin (1997)
Deza, M., Grishukhin, V.: Hypermetric graphs. Quart. J. Math. Oxford. 44(2), 399–433 (1993)
Deza, M., Grishukhin, V., Shtogrin, M.: Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and Zn. Imperial College Press, Berlin (2004)
Deza, M., Shpectorov, S.: Polyhexes that are \(l_1\)-graphs. Eur. J. Combin. 30, 1090–1100 (2009)
Djokovič, D.Ž: Distance-preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14, 263–267 (1973)
Marcusanu M. C.: The classification of \(l_1\)-embeddable fullerenes. PhD thesis, Bowling Green State University (2007)
Shpectorov, S.V.: On scale embeddings of graphs into hypercubes. Eur. J. Combin. 14, 117–130 (1993)
Tylkin, M.E.: On hamming geometry of unitary cubes. Doklady Akademii Nauk. 134, 1037–1040 (1960)
Wang, G., Zhang, H.: \(l_1\)-embeddability of hexagonal and quadrilateral mobius graphs. Ars Combin. 102, 269–287 (2011)
Wang, G., Zhang, H.: \(l_1\)-embeddability under the edge-gluing operation on graphs. Discrete Math. 313, 2115–2118 (2013)
Wang, G., Shpectorov, S.: \(l_1\)-embeddability of generic quadrilateral Möbius maps. Eur. J. Combin. 80, 373–389 (2019)
Zhang, H., Wang, G.: Embeddability of open-ended carbon nanotubes in hypercubes. Comp. Geom.-Theor. Appl. 43, 524–534 (2010)
Zhang, H., Xu, S.: None of the coronoid systems can be isometrically embedded into a hypercube. Discrete Appl. Math. 156, 2817–2822 (2008)
Acknowledgements
We would like to thank the reviewers’ comments concerning our manuscript. Those comments, besides the proof of the main theorem, are all valuable and very helpful for revising and improving our paper.
Funding
This work was supported by National Natural Science Foundation of China (Grant numbers [11861032] and [11961026], Natural Science Foundation of Jiangxi (Grant Number [20202BABL201010]). The authors have no relevant financial or non-financial interests to disclose. All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by [Guangfu Wang], [Zhikun Xiong] and [Lijun Chen]. The first draft of the manuscript was written by [Zhikun Xiong] and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by NSFC (Grant Nos.11861032, 11961026), Natural Science Foundation of Jiangxi(Grant No.20202BABL201010).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, G., Xiong, Z. & Chen, L. \(l_{1}\)-embeddability of shifted quadrilateral cylinder graphs. Graphs and Combinatorics 39, 129 (2023). https://doi.org/10.1007/s00373-023-02725-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-023-02725-w