Related papers: The graphs behind Reuleaux polyhedra

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…

A self-dual map $G$ is said to be \emph{antipodally self-dual} if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be…

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic…

Given $r \geq 3$, we prove that there exists $\lambda >0$ depending only on $r$ so that if $G$ is a metric graph of rank $r$ with metric entropy $1$, then there exists a proper subgraph $H$ of $G$ with metric entropy at least $\lambda$.…

We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the…

A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been…

For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to $\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For…

For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are…

Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider…

A perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a…

A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices.…

Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $\rho$, every graph $G$ with sufficiently large $\rho(G)$ contains a `well-structured' induced subgraph $H$ with…

Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\operatorname{C}(p,r,s)$, for any odd prime…

For any metric $d$ on $\mathbb{R}^2$, an ($\mathbb{R}^2,d$)-geometric graph is a graph whose vertices are points in $\mathbb{R}^2$, and two vertices are adjacent if and only if their distance is at most 1. If $d=\|.\|_{\infty}$, the metric…

A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded…

Given a graph $G$, the graph $[G]$ obtained by adding, for each pair of vertices of $G$, a unique vertex adjacent to both vertices is called the binding graph of $G$. In this work, we show that the class of binding graphs is…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu…

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of…