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Proper connection and proper-walk connection of digraphs

Author

Listed:
  • Fiedorowicz, Anna
  • Sidorowicz, Elżbieta
  • Sopena, Éric

Abstract

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the digraph D contains a directed path (a directed walk) from u to v such that arcs adjacent on that path (on that walk) have distinct colors. The proper connection number pc→(D) (the proper-walk connection number wc→(D)) of a digraph D is the minimum number of colours to make D properly connected (properly-walk connected). We prove that pc→(Cn(S))≤2 for every circulant digraph Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Furthermore, we give some sufficient conditions for a Hamiltonian digraph D to satisfy pc→(D)=wc→(D)=2.

Suggested Citation

  • Fiedorowicz, Anna & Sidorowicz, Elżbieta & Sopena, Éric, 2021. "Proper connection and proper-walk connection of digraphs," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    • Handle: RePEc:eee:apmaco:v:410:y:2021:i:c:s009630032100343x
    DOI: 10.1016/j.amc.2021.126253
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    References listed on IDEAS

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    1. Gu, Ran & Deng, Bo & Li, Rui, 2019. "Note on directed proper connection number of a random graph," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 169-174.
    2. Hui Lei & Shasha Li & Henry Liu & Yongtang Shi, 2018. "Rainbow vertex connection of digraphs," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 86-107, January.
    3. Wayne Goddard & Robert Melville, 2018. "Properly colored trails, paths, and bridges," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 463-472, February.
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    Citations

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    Cited by:

    1. Li, Zhenzhen & Wu, Baoyindureng, 2022. "Proper-walk connection of hamiltonian digraphs," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    2. Nie, Kairui & Ma, Yingbin & Sidorowicz, Elżbieta, 2023. "(Strong) Proper vertex connection of some digraphs," Applied Mathematics and Computation, Elsevier, vol. 458(C).

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