Related papers: Obtaining the Chamanara Surface from the van der C…

We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set…

Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of…

We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…

We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We…

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its…

Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the…

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant…

We describe a 3-parametric family $\mathcal{K}$ of properly embedded minimal tori with four parallel ends in quotients of $\mathbb{R}^3$ by two independent translations, which we will call the \textit{Standard Examples.} These surfaces…

A known Kronecker construction of completely regular codes has been investigated taking different alphabets in the component codes. This approach is also connected with lifting constructions of completely regular codes. We obtain several…

We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of…

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool…

We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive…

In his PhD Thesis, E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L.…

We introduce a family of graphs, which we call down-left graphs, and study their combinatorial and algebraic properties. We show that members of this family are well-covered, $C_5$-free, and vertex decomposable. By applying a result of…

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie…

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially…

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such…

We show that a cubic graph $G$ of order $n$ has an induced $2$-regular subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no induced cycle of…