Improper colouring of graphs with no odd clique minor
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AbstractAs a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no oddKtminor is (t− 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for eacht⩾ 2, every graph with no oddKtminor has a partition of its vertex set into 6t− 9 setsV1, …,V6t−9such that eachViinduces a subgraph of bounded maximum degree. Secondly, we prove that for eacht⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t−13 setsV1,…,V10t−13such that eachViinduces 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 496tsuch sets.
2015 ◽
Vol 114
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pp. 247-249
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2009 ◽
Vol 19
(02)
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pp. 119-140
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2016 ◽
Vol 09
(01)
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pp. 1650013
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2016 ◽
Vol 87
(2)
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pp. 337-341
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2017 ◽
Vol 31
(3)
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pp. 1572-1580
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