We propose local versions of Hadwiger's Conjecture, where only balls of radius $\mathrm{\Omega }(\log (v(G)))$ 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 $t\le 5$, 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 $13\cdot \max \{h(t),\lceil \frac{31}{2}(t-1)\rceil \})$ 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.

我们提出了哈维格猜想的本地版本，其中只有半径的球$\mathrm{\Omega }（\mathrm{日志}（v（G）））$每个顶点周围都需要$K_t$-无未成年人。我们问：如果一个图是局部的-$K_t$-无未成年人，可着色吗？我们证明答案是肯定的$t\le 5$，即使在更强的列表着色设置中，我们用$氧（\mathrm{日志}v（G））$-LOCAL模型中的圆形分布式着色算法。此外，我们表明，对于足够大的t值，我们可以在本地列出颜色-$K_t$-无次要图表$13\cdot \mathrm{最大限度}\{H（t）,\lceil \frac{31}{2}（t-1）\rceil \}）$颜色，哪里$H（t）$是任何值使得所有$K_t$-无次要图是$H（t）$-列表可着色。我们再次补充这一点$氧（\mathrm{日志}v（G））$-轮分布式算法。