Related papers: When does the table theorem imply a solution to th…

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…

We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph.…

In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky…

The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the…

In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.

We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows.…

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such…

Tangle-tree theorems are an important tool in structural graph theory, and abstract separation systems are a very general setting in which tangle-tree theorems can still be formulated and proven. For infinite abstract separation systems, so…

The cyclohedron (Bott-Taubes polytope) arises both as the polyhedral realization of the poset of all cyclic bracketings of a circular word and as an essential part of the Fulton-MacPherson compactification of the configuration space of n…

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n…

Here I present the present the first major result of a novel form of network analysis - a temporal interpretation. Treating numerical edges labels as the time at which an interaction occurs between the two vertices comprising that edge…

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.

Let F be a quasi-linear map on a separable normed space X, and assume that F splits on an infinite-dimensional subspace of X. Then the twisted sum topology induced by F on the direct sum of X and the real line can be written as the supremum…

In this paper we define a construct called a time-graph. A complete time-graph of order n is the cartesian product of a complete graph with n vertices and a linear graph with n vertices. A time-graph of order n is given by a subset of the…

We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning conic sections.

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…

We develop a formalism to address statistical pattern recognition of graph valued data. Of particular interest is the case of all graphs having the same number of uniquely labeled vertices. When the vertex labels are latent, such graphs are…

We connect and solve two longstanding open problems in quite different areas: the model-theoretic question of whether $SOP_2$ is maximal in Keisler's order, and the question from set theory/general topology of whether $\mathfrak{p} =…

We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated…