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Choosability with union separation of planar graphs without cycles of length 4

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  • Hou, Jianfeng
  • Zhu, Hongguo

Abstract

For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v ∈ V(G), |L(v)| ≥ k. Let s be a nonnegative integer. Then L is a (k,k+s)-list assignment of G if |L(u)∪L(v)|≥k+s for each edge uv. If for each (k,k+s)-list assignment L of G, G admits a proper coloring φ such that φ(v) ∈ L(v) for each v ∈ V(G), then we say G is (k,k+s)-choosable. This refinement of choosability is called choosability with union separation by Kumbhat, Moss and Stolee, who showed that all planar graphs are (3, 11)-choosable. In this paper, we prove that every planar graph without cycles of length 4 is (3,6)-choosable.

Suggested Citation

  • 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).
  • Handle: RePEc:eee:apmaco:v:386:y:2020:i:c:s0096300320304367
    DOI: 10.1016/j.amc.2020.125477
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    References listed on IDEAS

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    1. 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.
    2. Min Chen & Yingying Fan & Yiqiao Wang & Weifan Wang, 2017. "A sufficient condition for planar graphs to be (3, 1)-choosable," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 987-1011, November.
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    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, 2023. "On (2,r)-choosability of planar graphs without short cycles," Applied Mathematics and Computation, Elsevier, vol. 444(C).

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    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. Zhang, Ganchao & Chen, Min & Wang, Weifan, 2024. "A sufficient condition for planar graphs with girth 5 to be (1,6)-colorable," Applied Mathematics and Computation, Elsevier, vol. 474(C).

    More about this item

    Keywords

    Choosability; Planar graph; C4-free; Discharge;
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