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Improper colouring of graphs with no odd clique minor

  • Dong Yeap Kang (a1) and Sang-Il Oum (a1) (a2)

Abstract

As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

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Corresponding author

*Corresponding author. Email: sangil@kaist.edu

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The first author was supported by a TJ Park Science Fellowship of POSCO TJ Park Foundation.

Supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (no. NRF-2017R1A2B4005020).

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References

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[1] Bollobás, B. and Thomason, A. (1998) Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs. European J. Combin. 19 883887.
[2] Catlin, P. A. (1979) Hajós’ graph-coloring conjecture: variations and counterexamples. J. Combin. Theory Ser. B 26 268274.
[3] Cowen, L., Goddard, W. and Jesurum, C. E. (1997) Defective coloring revisited. J. Graph Theory 24 205219.
[4] Diestel, R. (2010) Graph Theory, fourth edition, Vol. 173 of Graduate Texts in Mathematics, Springer.
[5] Dvořák, Z. and Norin, S. (2017) Islands in minor-closed classes, I: Bounded treewidth and separators. arXiv:1710.02727
[6] Edwards, K., Kang, D. Y., Kim, J., Oum, S. and Seymour, P. (2015) A relative of Hadwiger’s conjecture. SIAM J. Discrete Math. 29 23852388.
[7] Esperet, L. and Joret, G. (2014) Colouring planar graphs with three colours and no large monochromatic components. Combin. Probab. Comput. 23 551570.
[8] Geelen, J., Gerards, B., Reed, B., Seymour, P. and Vetta, A. (2009) On the odd-minor variant of Hadwiger’s conjecture. J. Combin. Theory Ser. B 99 2029.
[9] Geelen, J. and Huynh, T. (2004) Colouring graphs with no odd-Kn minor. Manuscript. http://www.math.uwaterloo.ca/~jfgeelen/Publications/colour.pdf
[10] Guenin, B. (2005) Odd-K5-free graphs are 4-colourable. In Oberwolfach Report no. 3/2005, pp. 176178. https://www.mfo.de/document/0503/OWR_2005_03.pdf
[11] Hadwiger, H. (1943) Über eine Klassifikation der Streckenkomplexe. Vierteljschr. Naturforsch. Ges. Zürich 88 133142.
[12] Harary, F. (1953–1954) On the notion of balance of a signed graph. Michigan Math. J. 2 143146.
[13] van den Heuvel, J. and Wood, D. R. (2018) Improper colourings inspired by Hadwiger’s conjecture. J. London Math. Soc. 98 129148.
[14] Huynh, T., Oum, S. and Verdian-Rizi, M. (2017) Even-cycle decompositions of graphs with no odd-K4-minor. European J. Combin. 65 114.
[15] Jensen, T. R. and Toft, B. (1995) Graph Coloring Problems, Wiley.
[16] Kang, R. J. (2013) Improper choosability and Property B. J. Graph Theory 73 342353.
[17] Kawarabayashi, K.-I. (2008) A weakening of the odd Hadwiger’s conjecture. Combin. Probab. Comput. 17 815821.
[18] Kawarabayashi, K.-I. andMohar, B. (2007) A relaxed Hadwiger’s conjecture for list colorings. J. Combin. Theory Ser. B 97 647651.
[19] Komlós, J. and Szemerédi, E. (1996) Topological cliques in graphs, II. Combin. Probab. Comput. 5 7990.
[20] Kostochka, A. V. (1982) The minimum Hadwiger number for graphs with a given mean degree of vertices. Metody Diskret. Analiz. 38 3758.
[21] Kostochka, A. V. (1984) Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica 4 307316.
[22] Liu, C.-H. andOum, S. (2018) Partitioning H-minor free graphs into three subgraphs with no large components. J. Combin. Theory Ser. B 128 114133.
[23] Mohar, B., Reed, B. and Wood, D. R. (2017) Colourings with bounded monochromatic components in graphs of given circumference. Australas. J. Combin. 69 236242.
[24] Norin, S., Scott, A., Seymour, P. andWood, D. R. (2017) Clustered colouring in minor-closed classes. arXiv:1708.02370
[25] Ossona de Mendez, P., Oum, S. andWood, D. R. (2018) Defective colouring of graphs excluding a subgraph or minor. Combinatorica. doi: https://doi.org/10.1007/s00493-018-3733-1.
[26] Robertson, N., Seymour, P. and Thomas, R. (1993) Hadwiger’s conjecture for K6-free graphs. Combinatorica 13 279361.
[27] Seymour, P. (2016) Hadwiger’s conjecture. In Open Problems in Mathematics (Nash, J. and Rassias, M., eds), Springer.
[28] Thomason, A. (1984) An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95 261265.
[29] Thomason, A. (2001) The extremal function for complete minors. J. Combin. Theory Ser. B 81 318338.
[30] Wood, D. R. (2010) Contractibility and the Hadwiger conjecture. European J. Combin. 31 21022109.
[31] Wood, D. R. (2018) Defective and clustered graph colouring. Electron. J. Combin. #DS23.

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Improper colouring of graphs with no odd clique minor

  • Dong Yeap Kang (a1) and Sang-Il Oum (a1) (a2)

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