Coloring hypergraphs with excluded minors

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.Comment: 15 pages, corrected proof of Proposition

Linked Graphs with Restricted Lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x1, x2,..., xk, y1, y2,..., yk of distinct vertices, G contains k vertex-disjoint paths P1, P2,..., Pk such that Pi joins xi and yi for i = 1, 2,..., k. Moreover, the above defined k-linked graph G is k-linked modulo (m1, m2,..., mk) if, in addition, for any ktuple (d1, d2,..., dk) of natural numbers, the paths P1, P2,..., Pk can be chosen such that Pi has length di modulo mi for i = 1, 2,..., k. Thomassen showed that there exists a function f(m1, m2,..., mk) such that every f(m1, m2,..., mk)-connected graph is k-linked modulo (m1, m2,..., mk) provided all mi are odd. For even moduli, he showed in another article that there exists a natural number g(2, 2, · · · , 2) such that every g(2, 2, · · · , 2)-connected graph is k-linked modulo (2, 2, · · · , 2) if deleting any 4k − 3 vertices leaves a nonbipartite graph. In this paper, we give linear upper bounds for f(m1, m2,..., mk) and g(m1, m2,..., mk) in terms of m1, m2,..., mk, respectively. More specifically, we prove the following two results: (i) For any k-tuple (m1, m2,..., mk) of odd positiv