Local Hadwiger's Conjecture

TL;DR

This paper introduces local variants of Hadwiger's Conjecture, demonstrating that locally-$K_t$-minor-free graphs are t-colourable for small t and providing distributed algorithms for colouring such graphs.

Contribution

It establishes local conditions under which graphs are t-colourable and develops distributed algorithms for colouring graphs with local minor constraints.

Findings

Graphs locally-$K_t$-minor-free are t-colourable for t ≤ 5.

Distributed colouring algorithms operate in O(log v(G)) rounds.

Provides colour bounds for large t with distributed algorithms.

Abstract

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