Graphs and Matroids - Agnes Totschnig-Colouring graphs with forbidden 7-vertex minors

Abstract: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.