Abstract
Let G be a graph with vertex set V(G) and edge set E(G), and let f be an integer-valued function defined on V(G). It is proved in this paper that every bipartite \((0,mf-m+1)\)-graph has a (0, f)-factorization randomly r-orthogonal to n vertex-disjoint mr-subgraphs of G, which is a generalization of the known result with \(n=1\) given by Zhou and Wu.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akiyama, J., Kano, M.: Factors and factorizations of graphs- a survey. J. Graph Theory 9, 1–42 (1985)
Alspach, B., Heinrich, K., Liu, G.: Contemporary Design Theory-A Collection of Surveys. Wiley, New York (1992)
Feng, H., Liu, G.: Orthogonal factorizations of graphs. J. Graph Theory 40, 267–276 (2002)
Li, G., Chen, C., Yu, G.: Orthogonal factorizations of graphs. Discrete Math. 245, 173–194 (2002)
Li, G., Liu, G.: \((g, f)\)-factorizations orthogonal to a subgraph in graphs. Sci. China Ser. A 41, 267–272 (1998)
Liu, G.: Orthogonal \((g, f)\)-factorizations in graphs. Discrete Math. 143, 153–158 (1995)
Liu, G., Zhu, B.: Some problems on factorizations with constrains in bipartite graphs. Discrete Appl. Math. 128, 421–434 (2003)
Wang, C.: Subgraphs with orthogonal factorizations and algorithms. European J. Combin. 31, 1706–1713 (2010)
Zhou, S., Liu, H.: Discussions on orthogonal factorizations in digraphs. Acta Math. Appl. Sin. Engl. Ser. 38(2), 417–425 (2022)
Zhou, S., Liu, H., Zhang, T.: Randomly orthogonal factorizations with constraints in bipartite networks. Chaos Solitons Fract. 112, 31–35 (2018)
Zhou, S., Wu, J.: Randomly orthogonal factorizations of bipartite graphs. Rocky Mt. J. Math. 41(1), 337–348 (2011)
Zhou, S., Xu, Y., Sun, Z.: Degree conditions for fractional \((a, b, k)\)-critical covered graphs. Inform. Process. Lett. 152, 105838 (2019)
Zhou, S., Xu, L., Xu, Y.: Randomly orthogonal factorizations in networks. Chaos Solitons Fract. 93, 187–193 (2016)
Zhou, S., Zhang, T., Xu, Z.: Subgraphs with orthogonal factorizations in graphs. Discrete Appl. Math. 286, 29–34 (2020)
Acknowledgements
The authors express their sincere thanks to the editor and the anonymous referees for their valuable suggestions which greatly improved the original manuscript. The corresponding authors are Yuan Yuan (Email: yyuan@hainanu.edu.cn) and Rong-Xia Hao (Email: rxhao@bjtu.edu.cn).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare no competing financial 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 Hainan Provincial Natural Science Foundation of China(No. 120QN176) and the National Natural Science Foundation of China(Nos. 11971054, 11731002).
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
Yuan, Y., Hao, RX. Randomly r-orthogonal factorizations in bipartite graphs. Aequat. Math. 97, 511–522 (2023). https://doi.org/10.1007/s00010-022-00927-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-022-00927-w