IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v362y2019ic16.html
   My bibliography  Save this article

Remarks on the joins of 1-planar graphs

Author

Listed:
  • Ouyang, Zhangdong
  • Ge, Jun
  • Chen, Yichao

Abstract

A graph is called NIC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two pairs of crossing edges share at most one vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing each edge. It is known that any 1-planar (NIC-planar) graph of order n has at most 4n−8 (3.6n−7.2) edges and that this bound is tight. In this paper, we show that every 1-planar (NIC-planar) graph of order n with a dominating vertex has at most 4n−9 (3.5n−7.5) edges and this bound is tight for infinitely many values of n. Moreover, we derive tight upper bounds on the density of G, if G+2P1 and G+P2 are 1-planar (NIC-planar), respectively. These results improve some previous results due to Czap et al. and partially answer their open problem in negative. In addition, we also provide a complete characterization of outer 1-planarity of join of two graphs.

Suggested Citation

  • Ouyang, Zhangdong & Ge, Jun & Chen, Yichao, 2019. "Remarks on the joins of 1-planar graphs," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    • Handle: RePEc:eee:apmaco:v:362:y:2019:i:c:16
    DOI: 10.1016/j.amc.2019.06.051
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300319305168
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2019.06.051?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.

    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:362:y:2019:i:c:16. 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.