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Hadwiger’s conjecture for uncountable graphs

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Abstract

We consider the infinite form of Hadwiger’s conjecture. We give a(n apparently novel) proof of Halin’s 1967 theorem stating that every graph X with coloring number \(>\kappa \) (specifically with chromatic number \(>\kappa \)) contains a subdivision of \(K_\kappa \). We also prove that there is a graph of cardinality \(2^\kappa \) and chromatic number \(\kappa ^+\) which does not contain \(K_{\kappa ^+}\) as a minor. Further, it is consistent that every graph of size and chromatic number \(\aleph _1\) contains a subdivision of \(K_{\aleph _1}\).

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Acknowledgements

The author is grateful to one of the editors who quickly communicated an interesting and detailed history of the topic.

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  1. Institute of Mathematics, Eötvös University, Pázmány P. s. 1/C, Budapest, 1117, Hungary

    Péter Komjáth

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  1. Péter Komjáth
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Correspondence to Péter Komjáth.

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Komjáth, P. Hadwiger’s conjecture for uncountable graphs. Abh. Math. Semin. Univ. Hambg. 87, 337–341 (2017). https://doi.org/10.1007/s12188-016-0170-1

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  • DOI: https://doi.org/10.1007/s12188-016-0170-1

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