Related papers: Spotted disk and sphere graphs

For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

If $G$ is a graph then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$ where $d$ is the distance function. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an isometric…

In this paper, we first study what we call Superset-Subset-Disjoint (SSD) set system. Based on properties of SSD set system, we derive the following (I) to (IV): (I) For a nonnegative integer $k$ and a graph $G=(V,E)$ with $|V|\ge2$, let…

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into…

Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common. Let $S$ be a convex point set of $2n \geq 10$ points and let $\mathcal{H}$ be a family of…

The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appear in various places and found applications in many areas of mathematics including computational geometry,…

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is…

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound…

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at…

We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology.…

A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2.…

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a…

For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a…

A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an…

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge…

A multiarc and curve graph is a simplicial graph whose vertices are arc and curve systems on a compact, connected, orientable surface S. We show that all connected, non-trivial multiarc and curve graphs preserved by the natural action of…

Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or…

We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface. Previously, such decompositions had only been known for only a few complete graphs. These…

We introduce two common divisor graphs associated with a finite skew brace, based on its $\lambda$- and $\theta$-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most…