Abstract
Let n, k, t be positive integers. What is the maximum number of arcs in a digraph on n vertices in which there are at most t distinct walks of length k with the same endpoints? Denote \(f(t)=\max \{2t+1,2\left\lceil \sqrt{2t+9/4}+1/2\right\rceil +3\}\). In this paper, we prove that the maximum number is equal to \(n(n-1)/2\) and the extremal digraph are the transitive tournaments when \(k\ge n-1\ge f(t)\). Based on this result, we may determine the maximum numbers and the extremal digraphs when \(k\ge f(t)\) and n is sufficiently large, which generalizes the existing results. A conjecture is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material
Not applicable.
Code availability
Not applicable.
References
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. The Macmillan Press, London (1976)
Brown, W.G., Harary, F.: Extremal digraphs, combinatorial theory and its applications. Colloq. Math. Soc. Janos Bolyai 4(I), 135–198 (1970)
Brown, W.G., Erdős, P., Simonovits, M.: Extremal problems for directed graphs. J. Combin. Theory Ser. B 15, 77–93 (1973)
Brown, W.G., Simonovits, M.: Extremal Multigraph and Digraph Problems, Paul Erdős and His Mathematics, II (Budapest, 1999), pp. 157–203. Bolyai Soc. Math. Stud., 11, Jnos Bolyai Math. Soc., Budapest (2002)
Huang, Z., Lyu, Z.: 0–1 matrices whose \(k\)-th powers have bounded entries. Linear Multilinear Algebra 68, 1972–1982 (2020)
Huang, Z., Lyu, Z.: Extremal digraphs avoiding distinct walks of length 3 with the same endpoints. arXiv:2106.00212
Huang, Z., Lyu, Z., Qiao, P.: A Turán problem on digraphs avoiding distinct walks of a given length with the same endpoints. Discrete Math. 342, 1703–1717 (2019)
Huang, Z., Zhan, X.: Digraphs that have at most one walk of a given length with the same endpoints. Discrete Math. 311, 70–79 (2011)
Lyu, Z.: 0–1 matrices whose squares have bounded entries. Linear Algebra Appl. 607, 1–8 (2020)
Lyu, Z.: Digraphs that contain atmost \(t\) distinct walks of a given length with the same endpoints. J. Comb. Optim. 41, 762–779 (2021)
Lyu, Z.: Extremal digraphs avoiding distinct walks of length 4 with the same endpoints. Discuss. Math. Graph. Theory. https://doi.org/10.7151/dmgt.2321
Wu, H.: On the 0–1 matrices whose squares are 0–1 matrices. Linear Algebra Appl. 432, 2909–2924 (2010)
Zhan, X.: Matrix Theory. In: Graduate Studies in Mathematics 147. American Mathematical Society, Providence (2013)
Funding
This work was supported by the National Natural Science Foundation of China (No. 12171323) and the LiaoNing Revitalization Talents Program(XLYC2002017).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest/Competing interests
All author declares that he has no conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lyu, Z. A Note on Extremal Digraphs Containing at Most t Walks of Length k with the Same Endpoints. Graphs and Combinatorics 38, 33 (2022). https://doi.org/10.1007/s00373-021-02449-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-021-02449-9