Local Hadwiger's Conjecture

Abstract

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.