IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v33y2017i3d10.1007_s10878-016-0010-3.html
   My bibliography  Save this article

A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable

Author

Listed:
  • Miao Zhang

    (Zhejiang Normal University)

  • Min Chen

    (Zhejiang Normal University)

  • Yiqiao Wang

    (Beijing University of Chinese Medicine)

Abstract

A graph G is $$(d_1, d_2)$$ ( d 1 , d 2 ) -colorable if its vertices can be partitioned into subsets $$V_1$$ V 1 and $$V_2$$ V 2 such that in $$G[V_1]$$ G [ V 1 ] every vertex has degree at most $$d_1$$ d 1 and in $$G[V_2]$$ G [ V 2 ] every vertex has degree at most $$d_2$$ d 2 . Let $$\mathcal {G}_5$$ G 5 denote the family of planar graphs with minimum cycle length at least 5. It is known that every graph in $$\mathcal {G}_5$$ G 5 is $$(d_1, d_2)$$ ( d 1 , d 2 ) -colorable, where $$(d_1, d_2)\in \{(2,6), (3,5),(4,4)\}$$ ( d 1 , d 2 ) ∈ { ( 2 , 6 ) , ( 3 , 5 ) , ( 4 , 4 ) } . We still do not know even if there is a finite positive d such that every graph in $$\mathcal {G}_5$$ G 5 is (1, d)-colorable. In this paper, we prove that every graph in $$\mathcal {G}_5$$ G 5 without adjacent 5-cycles is (1, 7)-colorable. This is a partial positive answer to a problem proposed by Choi and Raspaud that is every graph in $$\mathcal {G}_5\;(1, 7)$$ G 5 ( 1 , 7 ) -colorable?.

Suggested Citation

  • Miao Zhang & Min Chen & Yiqiao Wang, 2017. "A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 847-865, April.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0010-3
    DOI: 10.1007/s10878-016-0010-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-016-0010-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-016-0010-3?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.

    References listed on IDEAS

    as
    1. Lingji Xu & Zhengke Miao & Yingqian Wang, 2014. "Every planar graph with cycles of length neither 4 nor 5 is $$(1,1,0)$$ -colorable," Journal of Combinatorial Optimization, Springer, vol. 28(4), pages 774-786, November.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Hou, Jianfeng & Jin, Yindong & Li, Heng & Miao, Lianying & Zhao, Qian, 2023. "On L(p,q)-labelling of planar graphs without cycles of length four," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    2. Hou, Jianfeng & Zhu, Hongguo, 2020. "Choosability with union separation of planar graphs without cycles of length 4," Applied Mathematics and Computation, Elsevier, vol. 386(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      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:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0010-3. 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.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

      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.