Graphs on the Torus and Geometry of Numbers

AbstractWe show that if G is a graph embedded on the torus S and eaeh nonnullhomotopic closed curve on S intersects G at least r times, then G contains at least ⌊34r⌋ pairwise disjoint nonnulihomotopic circuits. The factor 34 is best possible. We prove this by showing the equivalence of this bound to a bound in the two-dimensional geometry of numbers. To show the equivalence, we study integer norms, i.e., norms || · || such that ||x|| is an integer for each integer vector x. In particular, we show that each integer norm in two dimensions has associated with it a graph embedded on the torus, and conversely

Boxicity of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

Box representations of embedded graphs

A $d$-box is the cartesian product of $d$ intervals of $\mathbb{R}$ and a $d$-box representation of a graph $G$ is a representation of $G$ as the intersection graph of a set of $d$-boxes in $\mathbb{R}^d$. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function $f$, such that in every graph of genus $g$, a set of at most $f(g)$ vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function $f$ can be made linear in $g$. Finally, we prove that for any proper minor-closed class $\mathcal{F}$, there is a constant $c(\mathcal{F})$ such that every graph of $\mathcal{F}$ without cycles of length less than $c(\mathcal{F})$ has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio

Circuits in graphs embedded on the torus

AbstractWe give a survey of some recent results on circuits in graphs embedded on the torus. Especially we focus on methods relating graphs embedded on the torus to integer polygons in the Euclidean plane

Algorithms for Length Spectra of Combinatorial Tori

Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order. In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity $n$ in time $O(n^2 \log \log n)$ so that, given a cycle with $\ell$ edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in $O(\ell+\log n)$ time. Moreover, given any positive integer $k$, the first $k$ values of its unmarked length spectrum can be computed in time $O(k \log n)$. Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.Comment: 33 pages, 16 figure