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Questions tagged [combinatorial-topology]

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4
votes
1answer
210 views

Understanding proof about chromatic number

Consider an undirected graph $K(n,k,i)$, with the all $k$-element subsets of $\{1,\dots,n\}$ as vertices, and two vertices connected by an edge if their sets intersect in less than $i$ elements. ...
7
votes
1answer
326 views

Kneser graph with overlap

Consider a graph with the vertices being all subsets of size $n$ of a set of size $2n$. Two vertices are connected if their overlap has size at most one. What is the chromatic number of this graph? ...
7
votes
1answer
193 views

Uniqueness theorem for conformal maping

Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$. Assume that each has only finitely many ...
0
votes
0answers
105 views

Number of polyhedra with N faces?

A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...
6
votes
1answer
170 views

Existence of a particular kind of polygonal subdivisions of surfaces

Let $\Sigma$ be a closed, compact, connected, oriented smooth $2$-manifold (in other words, a sphere or a torus with $g$ handles). We may draw smooth arcs on the surface in such a way that they cut it ...
2
votes
0answers
100 views

Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear inequalities....
1
vote
1answer
83 views

property R_\lambda

A space X has property $R_\lambda$ if every family of its clopen sets of cardinality $\lambda$ has a subfamily of cardinality $\lambda$ which is either linked or disjoint. The following result was ...
9
votes
3answers
690 views

Combinatorial distance between simplicial complexes

Let $K_1$ and $K_2$ be two simplicial complexes. I am seeking a measure of the distance between $K_1$ and $K_2$ when viewed as combinatorial objects. What I have in mind is something like this. ...
21
votes
4answers
2k views

Testing simplicial complexes for shellability

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consider anything with ...
23
votes
0answers
2k views

Are there lots of integer homology three-spheres?

The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and references)...