IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v72y2018i3d10.1007_s10898-018-0647-9.html
   My bibliography  Save this article

Optimal channel assignment and L(p, 1)-labeling

Author

Listed:
  • Junlei Zhu

    (Zhejiang Normal University
    Jiaxing University)

  • Yuehua Bu

    (Zhejiang Normal University
    Zhejiang Normal University Xingzhi College)

  • Miltiades P. Pardalos

    (University of Florida)

  • Hongwei Du

    (Harbin Institute of Technology Shenzhen Graduate School)

  • Huijuan Wang

    (Qingdao University)

  • Bin Liu

    (Ocean University of China)

Abstract

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function $$f:V(G)\rightarrow \{0,1,2,\ldots ,k\}$$ f : V ( G ) → { 0 , 1 , 2 , … , k } such that $$|f(u)-f(v)|\ge p$$ | f ( u ) - f ( v ) | ≥ p if $$d(u,v)=1$$ d ( u , v ) = 1 and $$|f(u)-f(v)|\ge 1$$ | f ( u ) - f ( v ) | ≥ 1 if $$d(u,v)=2$$ d ( u , v ) = 2 , where d(u, v) is the distance between the two vertices u and v in the graph. Denote $$\lambda _{p,1}^l(G)= \min \{k \mid G$$ λ p , 1 l ( G ) = min { k ∣ G has a list k-L(p, 1)-labeling $$\}$$ } . In this paper we show upper bounds $$\lambda _{1,1}^l(G)\le \Delta +9$$ λ 1 , 1 l ( G ) ≤ Δ + 9 and $$\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}$$ λ 2 , 1 l ( G ) ≤ max { Δ + 15 , 29 } for planar graphs G without 4- and 6-cycles, where $$\Delta $$ Δ is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.

Suggested Citation

  • Junlei Zhu & Yuehua Bu & Miltiades P. Pardalos & Hongwei Du & Huijuan Wang & Bin Liu, 2018. "Optimal channel assignment and L(p, 1)-labeling," Journal of Global Optimization, Springer, vol. 72(3), pages 539-552, November.
  • Handle: RePEc:spr:jglopt:v:72:y:2018:i:3:d:10.1007_s10898-018-0647-9
    DOI: 10.1007/s10898-018-0647-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-018-0647-9
    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/s10898-018-0647-9?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. Yuehua Bu & Xubo Zhu, 2012. "An optimal square coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 580-592, November.
    2. Wei Dong & Wensong Lin, 2016. "An improved bound on 2-distance coloring plane graphs with girth 5," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 645-655, August.
    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).

    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.
    1. Qiming Fang & Li Zhang, 2022. "Sharp upper bound of injective coloring of planar graphs with girth at least 5," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1161-1198, September.
    2. Hoang La & Mickael Montassier, 2022. "2-Distance list $$(\Delta +2)$$ ( Δ + 2 ) -coloring of planar graphs with girth at least 10," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1356-1375, September.
    3. Wei Dong & Baogang Xu, 2017. "2-Distance coloring of planar graphs with girth 5," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1302-1322, November.
    4. Wei Dong & Wensong Lin, 2016. "An improved bound on 2-distance coloring plane graphs with girth 5," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 645-655, August.
    5. Yu, Jiahao & Chen, Min & Wang, Weifan, 2023. "2-Distance choosability of planar graphs with a restriction for maximum degree," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    6. Dan Yi & Junlei Zhu & Lixia Feng & Jiaxin Wang & Mengyini Yang, 2019. "Optimal r-dynamic coloring of sparse graphs," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 545-555, August.
    7. Yuehua Bu & Xiaoyan Yan, 2015. "List 2-distance coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(4), pages 1180-1195, November.

    More about this item

    Keywords

    Planar graph; Cycle; Labeling;
    All these keywords.

    Statistics

    Access and download statistics

    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:jglopt:v:72:y:2018:i:3:d:10.1007_s10898-018-0647-9. 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.