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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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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).
    2. 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).
    3. 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).

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    1. 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).

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