Journal of Combinatorial Optimization

, Volume 33, Issue 3, pp 847–865 | Cite as

A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable



A graph G is \((d_1, d_2)\)-colorable if its vertices can be partitioned into subsets \(V_1\) and \(V_2\) such that in \(G[V_1]\) every vertex has degree at most \(d_1\) and in \(G[V_2]\) every vertex has degree at most \(d_2\). Let \(\mathcal {G}_5\) denote the family of planar graphs with minimum cycle length at least 5. It is known that every graph in \(\mathcal {G}_5\) is \((d_1, d_2)\)-colorable, where \((d_1, d_2)\in \{(2,6), (3,5),(4,4)\}\). We still do not know even if there is a finite positive d such that every graph in \(\mathcal {G}_5\) is (1, d)-colorable. In this paper, we prove that every graph in \(\mathcal {G}_5\) 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 \(\mathcal {G}_5\;(1, 7)\)-colorable?.


Planar graphs Improper coloring Discharging method 



The authors thank the referees for their careful reading and valuable suggestions. M. Chen is Research supported by NSFC (Nos. 11471293, 11271335, 11401535), ZJNSFC (No. LY14A010014). Wang is Research supported by NSFC (No. 11301035).


  1. Borodin OV, Ivanova AO, Montassier M, Ochem P, Raspaud A (2010) Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k. J Graph Theory 65(2):83–93MathSciNetCrossRefMATHGoogle Scholar
  2. Borodin OV, Kostochka AV (2011) Vertex decompositions of sparse graphs into an independent set and a subgraph of maximum degree at most 1. Sibirsk Math Zh 52(5):1004–1010MathSciNetMATHGoogle Scholar
  3. Borodin OV, Kostochka AV (2014) Defective 2-colorings of sparse graphs. J Combin Theory Ser B 104:72–80MathSciNetCrossRefMATHGoogle Scholar
  4. Borodin OV, Kostochka A, Yancey M (2013) On 1-improper 2-coloring of sparse graphs. Comb Math 313(22):2638–2649MathSciNetMATHGoogle Scholar
  5. Cowen L, Cowen R, Woodall D (1986) Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J Graph Theory 10:187–195MathSciNetCrossRefMATHGoogle Scholar
  6. Choi I, Raspaud A (2015) Planar graphs with girth at least 5 are \((3,5)\)-colorable. Discret Math 338:661–667MathSciNetCrossRefMATHGoogle Scholar
  7. Eaton N, Hull T (1999) Defective list colorings of planar graphs. Bull Inst Combin Appl 25:40MathSciNetMATHGoogle Scholar
  8. Havet F, Sereni J-S (2006) Improper choosability of graphs and maximum average degree. J Graph Theory 52:181–199MathSciNetCrossRefMATHGoogle Scholar
  9. Hill O, Smith D, Wang Y, Xu L, Yu G (2013) Planar graphs without 4-cycles or 5-cycles are (3,0,0)-colorable. Discret Math 313:2312–2317CrossRefMATHGoogle Scholar
  10. Hill O, Yu G (2013) A relaxation of Steinberg’s conjecture. SIAM J Discret Math 27:584–596MathSciNetCrossRefMATHGoogle Scholar
  11. Li H, Wang Y, Xu J (2014) Planar graphs with cycles of length neither 4 nor 7 are (3,0,0)-colorable. Discret Math 327:29–35MathSciNetCrossRefMATHGoogle Scholar
  12. Montassier M, Ochem P (2013) Near-colorings: non-colorable graphs and np-completness. submitted for publicationGoogle Scholar
  13. Šrekovski R (1999) List improper colourings of planar graphs. Comb Probab Comput 8:293–299MathSciNetCrossRefGoogle Scholar
  14. Wang Y, Xu J (2013) Planar graphs with cycles of length neither 4 nor 6 are (2,0,0)-colorable. Inform Process Lett 113:659–663MathSciNetCrossRefMATHGoogle Scholar
  15. Wang Y, Xu J (2014) Improper colorability of planar graphs without prescribed short cycles. Discret Math 322:5–14MathSciNetCrossRefMATHGoogle Scholar
  16. Xu B (2008) on \((3,1)^*\)-coloring of plane graphs. SIAM J Discret Math 23:205–220MathSciNetCrossRefMATHGoogle Scholar
  17. Xu L, Miao Z, Wang Y (2014) Every planar graph with cycles of length neither 4 nor 5 is (1,1,0)-colorable. J Comb Optim 28:774–786MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.School of ManagementBeijing University of Chinese MedicineBeijingChina

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