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Site: VertexPartition

Vertex-partitions of planar graphs


Let us consider the following classes of graphs:

Observe that: d1=F, D0=I, D1=S1=F1

Consider i classes of graphs G1, . . . , Gi. A (G1, . . . , Gi)-partition of a graph G is a vertex-partition into i sets V1,...,Vi such that, for all 1 <= j <= i, the graph G[Vj] induced by Vj belongs to Gj. For example, a (I,F,d2)-partition is a vertex-partition into three sets V1,V2,V3 such that G[V1] is an empty graph, G[V2] is a forest, and G[V3] is a 2-degenerate graph.


Results

GirthVertex-partitionsReferences
3(I,I,I,I)K. Appel K, W. Haken. Every Planar Map is Four Colorable Part I. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.
  K. Appel, W. Haken W, J. Koch. Every Planar Map is Four Colorable Part II. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.
 (I,F,F)O.V. Borodin. A proof of Grünbaum’s conjecture on the acyclic 5-colorability of planar graphs (russian). Dokl. Akad. Nauk SSSR, 231(1):18–20, 1976.
 (F2,F2,F2)K.S. Poh. On the linear vertex-arboricity of a plane graph. Journal of Graph Theory, 14(1):73–75, 1990.
 (F,d2)C. Thomassen. Decomposing a planar graph into degenerate graphs. Journal of Combinatorial Theory, Series B, 65(2):305–314, 1995.
 (I,d3)C. Thomassen. Decomposing a planar graph into an independent set and a 3-degenerate graph. Journal of Combinatorial Theory, Series B, 83(2):262–271, 2001.
4(I,I,I)H. Grötzsch. Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120, 1959.
 (F,F)Folklore
 (F,F5)François Dross, Mickael Montassier, Alexandre Pinlou. Partitioning a triangle-free planar graph into a forest and a forest of bounded degree. http://arxiv.org/pdf/1601.01523v1.pdf
 (I,d2)Folklore
5(I,F)O.V. Borodin, A.N. Glebov. On the partition of a planar graph of girth 5 into an empty and an acyclic subgraph (russian). Diskretnyi Analiz i Issledovanie Operatsii, 8(4):34–53, 2001.
 (D1,D10)H. Choi, I. Choi, J. Jeong, G. Suh. (1,k)-coloring of graphs with girth at least 5 on a surface. http://arxiv.org/pdf/1412.0344v1.pdf
 (D2,D6)O.V. Borodin, A.V. Kostochka. Defective 2-colorings of sparse graphs. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.
 (D3,D5)I. Choi and A. Raspaud. Planar graphs with girth at least 5 are (3,5)-colorable. Discrete Mathematics, 318(4):661–667, 2015.
 (D4,D4)F. Havet, J.-S. Sereni. Improper choosability of graphs and maximum average degree. Journal of Graph Theory, 52(3):181-199, 2006.
6(D1,D4)O.V. Borodin, A.V. Kostochka. Defective 2-colorings of sparse graphs. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.
 (I,Dk) for any kO.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k. Journal of Graph Theory, 65(2):83-93, 2010.
 (D2,D2)F. Havet, J.-S. Sereni. Improper choosability of graphs and maximum average degree. Journal of Graph Theory, 52(3):181-199, 2006.
 (P14,P14)M. Axenovich, T. Ueckerdt, and P. Weiner. Splitting planar graphs of girth 6 into two linear forests with short paths. http://arxiv.org/pdf/1507.02815.pdf
 (O12,O12)L. Esperet and P. Ochem. Islands in graphs on surfaces. http://arxiv.org/pdf/1402.2475v3.pdf
7(I,D4)O.V. Borodin, A.V. Kostochka. Defective 2-colorings of sparse graphs. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.
 (I,F5)F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree. http://arxiv.org/pdf/1606.04394v1.pdf
 (D1,D1)O.V. Borodin, A. Kostochka, M. Yancey. On 1-improper 2-coloring of sparse graphs. Discrete Mathematics, 313(22):2638-2649, 2013.
 (I,D2)M. Montassier and P. Ochem. Near-colorings: non-colorable graphs and NP-completness. Electronic Journal of Combinatorics,Volume 22, Issue 1, 2015.
8(I,D2)O.V. Borodin, A.V. Kostochka. Defective 2-colorings of sparse graphs. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.
 (I,F3)F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree. http://arxiv.org/pdf/1606.04394v1.pdf
9(I,D1)L. Esperet, M. Montassier, P. Ochem, and A. Pinlou. A complexity dichotomy for the coloring of sparse graphs. Journal of Graph Theory, 73(1):85-102, 2013.
10(I,D1) ?Open question.
 (I,F2)F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree. http://arxiv.org/pdf/1606.04394v1.pdf
11(I,D1)J. Kim, A. Kostochka, X. Zhu.Improper coloring of sparse graphs with a given girth, I: (0,1)-colorings of triangle-free graphs. European Journal of Combinatorics, 42:26-48, 2014.

Acknowledgements

Thanks to Ilkyoo Choi, François, Dross, Pascal Ochem.


Keywords: graph theory, graphs, partition, vertex, planar graphs, given girth

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