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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
4
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0
answers
129
views
What is the crossing number of dodecahedron with a copy of $K_5$ inside each face
Suppose we are given a regular dodecahedron. Then we add five crossed edges inside each of its faces (actually, inside each face it is a copy of $K_5$). It is clear that this drawing has 60 crossings. …
3
votes
1
answer
196
views
Looking for examples showing that the crossing number may not be realized by the drawings wi...
The crossing number $cr(G)$ of a graph $G$ is the lowest number of edge crossings of a plane drawing of the graph $G$. The local crossing number of a drawing of a graph is the largest number of crossi …
6
votes
2
answers
289
views
Is there any maximal 1-planar or 2-planar graph that is not 3-connected
A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. A $k$-planar graph $G$ is maximal if $G+uv$ is not $k$-planar for any non-adjacent vertices $u,v\i …
5
votes
1
answer
260
views
What is the crossing number of cube with a pair of crossing edges inside each face
Suppose we are given a cube and we add a pair of crossing edges inside each of its faces. It is clear that this drawing has 6 crossings. My question is whether such a graph has crossing number 6? How …
6
votes
1
answer
286
views
Generating 21-vertex 4-regular plane graphs with 8 faces of degree 3 and 15 faces of degree 4
Is there any way to generate all 4-regular plane graphs with 21 vertices, 8 faces of degree 3, and 15 faces of degree 4? If so, how many of these graphs are there and what are they?
3
votes
Accepted
Is there any maximal 1-planar or 2-planar graph that is not 3-connected
This is a non-3-connected 1-planar example...
7
votes
2
answers
251
views
There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....?
Is there any characterization on the set of integers $n$ such that there is a 3-connected 5-regular simple $n$-vertex planar graph?
4
votes
Is it possible that every edge in a 1-planar drawing with minimum number of crossings is cro...
Concerning the above answer, I do not quite agree with the last sentence. Why is there some vertex of $D^\times$ outside $O$ if $y$ is a dummy vertex on $C$?
I draw a figure, where $y$ is a dummy vert …