All Questions
Tagged with graph-drawing planar-graphs
14 questions
2
votes
0
answers
73
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Is the chromatic number of every 7-connected 1-planar graph at most 5?
1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors. Later, the precise number of colors needed to color these graphs, in the worst case, ...
- 1,851
1
vote
0
answers
97
views
Problem related to crossing number
Let $G$ be a graph embedded in the plane (with crossings). For $ F \subset E(G) $, denote by $c(F)$ the set of edges of $G$ that cross some edge in $F$.
Denote $\delta(v)$ the set of edges with one ...
- 111
1
vote
0
answers
77
views
Is there a variant of the crossing lemma for multigraphs with arbitrary embedding?
Suppose $G$ is a graph embedded in the plane with $m=|E(G)|$ edges and $n=|V(G)|$ vertices.
Suppose $\operatorname{sim}(G)$, the simplification of $G$ contains $ m' \gg 3n $ edges.
Call the set of ...
- 111
2
votes
0
answers
63
views
What is the range of connectivity for maximal IC-planar graphs?
A graph is IC-planar if it admits a drawing in the plane with
at most one crossing per edge and such that two pairs of crossing edges
share no common end vertex. A graph $G$ is maximal in a graph ...
- 1,851
1
vote
1
answer
173
views
Who introduced the concept of beyond planar graphs?
The concept of planar graphs seems to be standard (I'm also not sure who first used this term), and recently, beyond planar graphs attract a lot of interest in the field of graph drawing. I know that ...
- 1,851
0
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0
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81
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 ...
- 1,190
6
votes
3
answers
526
views
Enumerating all inequivalent planar embeddings of a planar graph
Graph $G$ can be embedded (or has an embedding) in the space if $G$ can be drawn in the space if $G$ can be drawn in such a way that no two edges cross except at an end-vertex in common. A Graph $G$ ...
- 1,851
3
votes
0
answers
136
views
Is there a more intuitive proof that a 1-planar graph with minimum degree 7 contains a $K_4$?
In the following paper, Hudák Dávid, and Tomáš Madaras give the following Theorem 1.1.
Hudák, Dávid, and Tomáš Madaras. "On local properties of 1-planar graphs with high minimum degree." ...
- 1,851
1
vote
1
answer
160
views
Can Tutte embedding be guaranteed that each face is convex?
In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a ...
- 1,851
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
10
votes
2
answers
599
views
Method to draw 3-connected planar graph on a sphere
The Tutte embedding is a way to create a "nice" drawing of a 3-connected planar graph in the plane, after having chosen an outer face.
Is there a similar method to draw such a graph on a sphere? ...
- 911
5
votes
2
answers
553
views
"Locally Nonplanar" graph
A 2-connected $3$-connected graph $G$ is "Almost Planar" Locally Nonplanar if it has a a $2$-connected spanning subgraph $H$ and an embedding in the plane such that $H$ is planar in this embedding and ...
- 1,034
5
votes
1
answer
479
views
Can all crossings in a graph be moved to one point?
Consider a graph $G$ with at least two unavoidable crossings, say, the disjoint union of two copies of $K_5$. Can such a graph always be drawn so that there is only one singular point (where all ...
- 4,829
7
votes
2
answers
1k
views
Planar layouts of bipartite graphs
Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar.
It would be very convenient if there was a ...
- 4,462