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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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    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.
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    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.

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