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In 1943, Hadwiger conjectured that every k-chromatic graph has a K_k-minor. While the cases k = 5 and k = 6 have been shown to be equivalent to the Four Colour Theorem, respectively by Wagner, and in seminal work by Robertson, Seymour and Thomas, the cases k at least 7 remain open. We show that any 7-chromatic graph has as a minor the complete graph K_7 with two adjacent edges removed, by extending work of Kawarabayashi and Toft and by proving a new edge-extremal bound. This improves Jakobsen’s result with two arbitrary edges removed.

Joint work with Sergey Norin.

(Agnes Totschnig) [MIT]

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Remarks / Remarques

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English
LaBRI/178