Local Hadwiger's Conjecture
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We propose local versions of Hadwiger's Conjecture, where only balls of radius Ω(log v(G)) around each vertex are required to be K_t-minor-free. We ask: if a graph is locally-K_t-minor-free, is it t-colourable? We show that the answer is yes when t ≤ 5, even in the stronger setting of list-colouring, and we complement this result with a O(log v(G))-round distributed colouring algorithm in the LOCAL model. Further, we show that for large enough values of t, we can list-colour locally-K_t-minor-free graphs with 13 ·max{h(t),⌈31/2(t-1) ⌉} colours, where h(t) is any value such that all K_t-minor-free graphs are h(t)-list-colourable. We again complement this with a O(log v(G))-round distributed algorithm.
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