IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v258y2015icp155-161.html
   My bibliography  Save this article

Note on the complexity of deciding the rainbow (vertex-) connectedness for bipartite graphs

Author

Listed:
  • Li, Shasha
  • Li, Xueliang
  • Shi, Yongtang

Abstract

A path in an edge-colored graph is said to be a rainbow path if no two edges on the path share the same color. An edge-colored graph is (strongly) rainbow connected if there exists a rainbow (geodesic) path between every pair of vertices. The (strong) rainbow connection number of G, denoted by (scr(G), respectively) rc(G), is the smallest number of colors that are needed in order to make G (strongly) rainbow connected. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. Though for a general graph G it is NP-Complete to decide whether rc(G)=2 (or rvc(G)=2), in this paper, we show that the problem becomes easy when G is a bipartite graph. Whereas deciding whether rc(G)=3 (or rvc(G)=3) is still NP-Complete, even when G is a bipartite graph. Moreover, it is known that deciding whether a given edge(vertex)-colored (with an unbound number of colors) graph is rainbow (vertex-) connected is NP-Complete. We will prove that it is still NP-Complete even when the edge(vertex)-colored graph is bipartite. We also show that a few NP-hard problems on rainbow connection are indeed NP-Complete.

Suggested Citation

  • Li, Shasha & Li, Xueliang & Shi, Yongtang, 2015. "Note on the complexity of deciding the rainbow (vertex-) connectedness for bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 155-161.
    • Handle: RePEc:eee:apmaco:v:258:y:2015:i:c:p:155-161
    DOI: 10.1016/j.amc.2015.02.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630031500168X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.02.015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Li, Jingjian & Xu, Shangjin & Cao, Mengyue & Kang, Zhe, 2017. "On tetravalent symmetric dihedrants," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 49-55.
    2. Li, Jing Jian & Ling, Bo & Liu, Guodong, 2018. "A characterisation on arc-transitive graphs of prime valency," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 227-233.
    3. Yongtang Shi & Meiqin Wei & Jun Yue & Yan Zhao, 2017. "Coupon coloring of some special graphs," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 156-164, January.
    4. Chen, Lin & Li, Xueliang & Liu, Jinfeng, 2017. "The k-proper index of graphs," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 57-63.
    5. Yue, Jun & Zhang, Shiliang & Zhang, Xia, 2016. "Note on the perfect EIC-graphs," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 481-485.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:258:y:2015:i:c:p:155-161. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.