Related papers: Connectivity of ample, conic and random simplicial…
A compact K\"ahler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective…
A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A…
A theorem due to Seyffarth states that every planar $4$-connected $n$-vertex graph has a cycle double cover (CDC) containing at most $n-1$ cycles (a "small" CDC). We extend this theorem by proving that, in fact, such a graph must contain…
In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the…
We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree…
The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this…
We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a…
For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at…
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$,…
In this note, we prove that every 4-connected optimal 2-planar graph is Hamiltonian-connected. Furthermore, we show that the 4-connectedness condition is sharp by constructing infinitely many 3-connected optimal 2-planar graphs that are…
Necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a 3-connected simple graph are detailed. Conditions are also given under which such a sequence is necessarily 3-connected i.e. the sequence…
Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$,…
The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a…
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected…
This article proves hypersurfaces of degree d in projective n-space are "rationally simply-connected" if $d^2 \leq n$. In a forthcoming paper, de Jong and I prove a slightly weaker result when $d^2 \leq n+1$.
An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is the maximum number of edges in an…
This paper concerns the facial geometry of the set of $n \times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r \leq \sqrt{\mathrm{c} n}$, where…
Given a finite set T of maps on a finite ring R, we look at the finite simple graph G=(V,E) with vertex set V=R and edge set E={(a,b) | exists t in T, b=t(a), b not equal to a}. An example is when R=Z_n and T consists of a finite set of…
An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…
We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. We give some applications of this method to complexes arising from graphs. As a consequence, we answer some questions raised in…