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Improper colouring of graphs with no odd clique minor

Part of: Graph theory

Published online by Cambridge University Press:  04 February 2019

Dong Yeap Kang  and
Sang-Il Oum
Dong Yeap Kang
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon, 34141South Korea
Sang-Il Oum*
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon, 34141South Korea Discrete Mathematics Group, Institute for Basic Science (IBS), 55 Expo-ro Yuseong-gu Daejeon, 34126South Korea
*
*Corresponding author. Email: sangil@kaist.edu
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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.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C83: Graph minors
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 740 - 754
Copyright
© Cambridge University Press 2019 

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Footnotes

†

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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