Coloring hypergraphs with excluded minors

TL;DR

This paper extends Hadwiger's conjecture from graphs to hypergraphs, establishing bounds on hypergraph coloring related to minors, and explores implications for graph coloring and minors in special classes.

Contribution

It introduces a hypergraph minor concept, proves a weak extension of Hadwiger's conjecture for hypergraphs, and provides bounds and conjectures for hypergraph coloring related to minors.

Findings

Hypergraph minors can be used to extend Hadwiger's conjecture.

Established bounds: (t) between eil(1.5(t-1)), 2g(t).

Conjecture: (t) = eil(1.5(t-1)) for all t.

Abstract

Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly $(t-1)$-colorable. The purpose of this work is to demonstrate that a natural extension of Hadwiger's problem to hypergraph coloring exists, and to derive some first partial results and applications. Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_1$ is a minor of a hypergraph $H_2$, if a hypergraph isomorphic to $H_1$ can be obtained from $H_2$ via a finite sequence of vertex- and hyperedge-deletions, and hyperedge contractions. We first show that a weak extension of Hadwiger's conjecture to hypergraphs holds true: For every $t \ge 1$, there exists a finite (smallest) integer $h(t)$ such that every hypergraph with no $K_t$-minor is $h(t)$-colorable, and we prove $$\left\lceil\frac{3}{2}(t-1)\right\rceil \le h(t) \le 2g(t)$$ where $g(t)$ denotes the maximum chromatic number of graphs with no $K_t$-minor. Using the recent result by Delcourt and Postle that $g(t)=O(t \log \log t)$, this yields $h(t)=O(t \log \log t)$. We further conjecture that $h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$, i.e., that every hypergraph with no $K_t$-minor is $\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable for all $t \ge 1$, and prove this conjecture for all hypergraphs with independence number at most $2$. By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as: -graphs of chromatic number $C k t \log \log t$ contain $K_t$-minors with $k$-edge-connected branch-sets, -graphs of chromatic number $C q t \log \log t$ contain $K_t$-minors with modulo-$q$-connected branch sets, -by considering cycle hypergraphs of digraphs we recover known results on strong minors in digraphs of large dichromatic number as special cases.