All Questions
7 questions
9
votes
3
answers
470
views
Is it possible that every edge in a 1-planar drawing with minimum number of crossings is crossed?
A graph is 1-planar is it has drawing in the plane so that each edge is crossed at most once. Here we also assume the drawing satisfies (1) no edge is self-crossed; (2) no two adjacent edges are ...
- 557
4
votes
0
answers
131
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. ...
- 1,190
5
votes
1
answer
263
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 ...
- 1,190
3
votes
1
answer
200
views
Looking for examples showing that the crossing number may not be realized by the drawings with local crossing number
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 ...
- 1,190
7
votes
2
answers
252
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?
- 1,190
6
votes
2
answers
293
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 $u,v\...
- 1,190
1
vote
1
answer
228
views
Counting edges in embeddable CW-complexes in R^3
Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...
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