Grid Minors of Graphs on the Torus

AbstractWe show that any graph G embedded on the torus with face-width r ≥ 5 contains the toroidal ⌊23r⌋-grid as a minor. (The face-width of G is the minimum value of |C∩G|, where C ranges over all homotopically nontrivial closed curves on the torus. The toroidal k-grid is the product Ck × Ck of two copies of a k-circuit Ck.) For each fixed r ≥ 5, the value ⌊23r⌋ is largest possible. This applies to a theorem of Robertson and Seymour showing, for each graph H embedded on any compact surface S, the existence of a number ρH such that every graph G embedded on S with face-width at least ρH contains H as a minor. Our result implies that for H = Ck × Ck embedded on torus, ρH ≔ ⌈32k⌉ is the smallest possible value. Our proof is based on deriving a result in the geometry of numbers. It implies that for any symmetric convex body K in R2 one has λ2(K)·λ1(K*) ≤ 32 and that this bound is smallest possible. (Here λi(K) denotes the minimum value of λ such that λ·K contains i linearly independent integer vectors. K* is the polar convex body.

Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

We find a graph of genus $5$ and its drawing on the orientable surface of genus $4$ with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus $4$. As a base step in the construction we use a counterexample to an extension of the unified Hanani-Tutte theorem on the torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem

Hexagonal Tilings and Locally C6 Graphs

We give a complete classification of hexagonal tilings and locally C6 graphs, by showing that each of them has a natural embedding in the torus or in the Klein bottle. We also show that locally grid graphs are minors of hexagonal tilings (and by duality of locally C6 graphs) by contraction of a perfect matching and deletion of the resulting parallel edges, in a form suitable for the study of their Tutte uniqueness.Comment: 14 figure

The Z2\mathbb{Z}_2-genus of Kuratowski minors

A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most $2$ common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler $\mathbb{Z}_2$-genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure