Related papers: K5 and K3,3 are Toroidal Penny Graphs

The Kuratowski graphs $K_{3,3}$ and $K_5$ characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by using Burnside's Lemma and their automorphism groups, without actually…

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on…

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over $k=\mathbb{F}_{p^2}$, that is optimal if $p=3$.

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the…

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super…

Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the…

Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T,…

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the…

In this paper, all finite groups whose commuting (non-commuting) graphs can be embed on the plane, torus or projective plane are classified.

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…

In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is…

In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…

We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by…

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a…

We study the distribution of algebraic points on K3 surfaces.

We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.

If a graph $G$ can be embedded on the torus, and be embedded linklessly in $\mathbb{R}^3$, it's not known whether or not we can always find a linkless embedding of $G$ contained in the standard (unknotted) torus; We show that, for orders 9…

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus $T^2 \subset \mathbb{R}^3$ and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not…

We give a computer-assisted proof of the fact that $R(K_5-P_3, K_5)=25$. This solves one of the three remaining open cases in Hendry's table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no…

In this paper, we show that if $G$ is $3$-connected, then the Cartesian product of graphs $G \square H$ embeds on the torus if and only if $G$ is outer-cylindrical and $H$ is a path on two vertices, $P_2$. As a by-product of our work, we…