IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v28y2014i1d10.1007_s10878-013-9680-2.html
   My bibliography  Save this article

Minimum number of disjoint linear forests covering a planar graph

Author

Listed:
  • Huijuan Wang

    (Shandong University)

  • Lidong Wu

    (University of Texas at Dallas)

  • Weili Wu

    (University of Texas at Dallas
    TaiYuan University of Technology)

  • Jianliang Wu

    (Shandong University)

Abstract

Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree $$\varDelta \ge 7$$ Δ ≥ 7 . We have proved that the linear arboricity of $$G$$ G is $$\lceil \frac{\varDelta }{2}\rceil $$ ⌈ Δ 2 ⌉ , if for each vertex $$v\in V(G)$$ v ∈ V ( G ) , there are two integers $$i_v,j_v\in \{3,4,5,6,7,8\}$$ i v , j v ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that any two cycles of length $$i_v$$ i v and $$j_v$$ j v , which contain $$v$$ v , are not adjacent. Clearly, if $$i_v=i, j_v=j$$ i v = i , j v = j for each vertex $$v\in V(G)$$ v ∈ V ( G ) , then we can easily get one corollary: for two fixed integers $$i,j\in \{3,4,5,6,7,8\}$$ i , j ∈ { 3 , 4 , 5 , 6 , 7 , 8 } , if there is no adjacent cycles with length $$i$$ i and $$j$$ j in $$G$$ G , then the linear arboricity of $$G$$ G is $$\lceil \frac{\varDelta }{2}\rceil $$ ⌈ Δ 2 ⌉ .

Suggested Citation

  • Huijuan Wang & Lidong Wu & Weili Wu & Jianliang Wu, 2014. "Minimum number of disjoint linear forests covering a planar graph," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 274-287, July.
  • Handle: RePEc:spr:jcomop:v:28:y:2014:i:1:d:10.1007_s10878-013-9680-2
    DOI: 10.1007/s10878-013-9680-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-013-9680-2
    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/s10878-013-9680-2?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. Weifan Wang & Yiqiao Wang, 2010. "Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree," Journal of Combinatorial Optimization, Springer, vol. 19(4), pages 471-485, May.
    2. Xiangwen Li & Vicky Mak-Hau & Sanming Zhou, 2013. "The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups," Journal of Combinatorial Optimization, Springer, vol. 25(4), pages 716-736, May.
    3. Patrizio Angelini & Fabrizio Frati, 2012. "Acyclically 3-colorable planar graphs," Journal of Combinatorial Optimization, Springer, vol. 24(2), pages 116-130, August.
    4. S. Bessy & F. Havet, 2013. "Enumerating the edge-colourings and total colourings of a regular graph," Journal of Combinatorial Optimization, Springer, vol. 25(4), pages 523-535, May.
    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. Huijuan Wang & Bin Liu & Xin Zhang & Lidong Wu & Weili Wu & Hongwei Gao, 2016. "List edge and list total coloring of planar graphs with maximum degree 8," Journal of Combinatorial Optimization, Springer, vol. 32(1), pages 188-197, July.

    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. Huijuan Wang & Lidong Wu & Xin Zhang & Weili Wu & Bin Liu, 2016. "A note on the minimum number of choosability of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1013-1022, April.
    2. Huijuan Wang & Bin Liu & Xin Zhang & Lidong Wu & Weili Wu & Hongwei Gao, 2016. "List edge and list total coloring of planar graphs with maximum degree 8," Journal of Combinatorial Optimization, Springer, vol. 32(1), pages 188-197, July.
    3. Chengchao Yan & Danjun Huang & Dong Chen & Weifan Wang, 2014. "Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five," Journal of Combinatorial Optimization, Springer, vol. 28(4), pages 893-909, November.
    4. Enqiang Zhu & Zepeng Li & Zehui Shao & Jin Xu & Chanjuan Liu, 2016. "Acyclic 3-coloring of generalized Petersen graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 902-911, February.
    5. Jingjing Huo & Yiqiao Wang & Weifan Wang, 2017. "Neighbor-sum-distinguishing edge choosability of subcubic graphs," Journal of Combinatorial Optimization, Springer, vol. 34(3), pages 742-759, October.
    6. Zehui Shao & Jin Xu & Roger K. Yeh, 2016. "$$L(2,1)$$ L ( 2 , 1 ) -labeling for brick product graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 447-462, February.
    7. Yi Wang & Jian Cheng & Rong Luo & Gregory Mulley, 2016. "Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 874-880, February.
    8. Junlei Zhu & Yuehua Bu & Yun Dai, 2018. "Upper bounds for adjacent vertex-distinguishing edge coloring," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 454-462, February.
    9. Huijuan Wang & Panos M. Pardalos & Bin Liu, 2019. "Optimal channel assignment with list-edge coloring," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 197-207, July.
    10. Hervé Hocquard & Mickaël Montassier, 2013. "Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 152-160, July.
    11. Joanna Skowronek-Kaziów, 2017. "Graphs with multiplicative vertex-coloring 2-edge-weightings," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 333-338, January.

    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:28:y:2014:i:1:d:10.1007_s10878-013-9680-2. 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.