Related papers: Complexity of atriodic continua
It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to…
A simplicial complex is a generalization of a graph: a collection of n-ary relationships (instead of binary as the edges of a graph), named simplices. In this paper, we develop a new tool to study the structure of simplicial complexes: we…
A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the…
We consider closed simplicial and cubical $n$-complexes in terms of link of their $(n-2)$-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every $(n-2)$-face is contained in 3 or 4 $n$-faces. Such simplicial…
We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…
We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting…
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable…
We show NP-completeness for several planar variants of the monotone satisfiability problem with bounded variable appearances. With one exception the presented variants have an associated bipartite graph where the vertex degree is bounded by…
When doing representation learning on data that lives on a known non-trivial manifold embedded in high dimensional space, it is natural to desire the encoder to be homeomorphic when restricted to the manifold, so that it is bijective and…
A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), it is a long-standing open problem to establish a tight upper bound…
We study a class of chainable continua which contains, among others, all inverse limit spaces generated by a single interval bonding map which is piecewise monotone and locally eventually onto. Such spaces are realized as attractors of…
Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $\phi:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $\phi(a_{1}... a_{n})=\phi(a_{1})...\phi(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this…
Let $\mathcal{A}$ be a set of positive numbers. A graph $G$ is called an $\mathcal{A}$-embeddable graph in $\mathbb{R}^d$ if the vertices of $G$ can be positioned in $\mathbb{R}^d$ so that the distance between endpoints of any edge is an…
We consider the bar complex of a monomial non-unital associative algebra $A=k \langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes $B_w$, defined for any fixed monomial $w=x_1...x_n \in A$. We give a simple argument,…
In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$…
In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and…
We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} \gamma^n n!$ for a constant $c_1>0$ and $\gamma \approx 11.89235$. We also determine the number of rooted simple chordal…
We study the complexity of S-adic sequences corresponding to a family of 216 multi-dimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main…
A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the…
For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and…