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Results tagged with graph-drawing
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user 148974
Problems related to graph drawing such as crossing numbers, layout designs, and intersection graphs.
5
votes
2
answers
249
views
Finding a special plane graph with some requirements on the faces
Is there a plane graph such that
(1) the outer face has degree 3, i.e, is a triangle,
(2) every inner face has degree 5, and
(3) any two degree 5 faces share at most one commong edge.
- 1,190
4
votes
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. …
- 1,190
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 …
- 1,190
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 …
- 1,190
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 …
- 1,190
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...
- 1,190
1
vote
1
answer
87
views
Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number …
- 1,190
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?
- 1,190
2
votes
1
answer
127
views
The density of a tripartite 1-planar graph
1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $n$-vertex 1-planar graph is at most $4n-8$, and the …
- 1,190
1
vote
Accepted
The density of a tripartite 1-planar graph
I find the following source. The known bound is 3.5n-7 (Theorem 4.8, pp. 57).
https://www.springer.com/gp/book/9789811565328
Beyond Planar Graphs
Communications of NII Shonan Meetings
Editors: Hong, S …
- 1,190
0
votes
0
answers
80
views
Is there is a constant $c$ such that toroidal graphs are minor-$c$-colorable?
A toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
A minor of graph G is a graph obtained from G by me …
- 1,190
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 …
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