All Questions
7 questions
7
votes
1
answer
300
views
The origin of a planar graph theorem of Steinitz and Rademacher
The subsequent statements are extracted from the article titled 'Generating r-regular graphs' (https://doi.org/10.1016/S0166-218X(02)00593-0).
A well-known classical theorem of Steinitz and ...
- 1,851
3
votes
1
answer
158
views
Sharp upper bound of the number of edges for graphs of thickness two
A graph $G=(V,E)$ has thickness $2$ if $E$ can be written as a disjoint union $E=E_1\cup E_2$ so that $G_1:=(V,E_1),G_2:=(V,E_2)$ are planar graphs. For instance, $K_5$ has thickness $2$. It is known ...
- 1,065
2
votes
0
answers
106
views
Decomposing a planar graph
Thomassen proved that the vertex set of every planar graph can be decomposed into two sets inducing a 1-degenerate graph and a 2-degenerate graph, respectively (C. Thomassen, Decomposing a planar ...
- 3,153
4
votes
1
answer
552
views
Product of vertex degrees of an edge in a planar graph
Let $G$ be a planar graph, which we may assume to be a triangulation, with vertex set $V$ and edge set $E$. Suppose the minimum vertex degree is at least 3, and suppose any two distinct edges share at ...
- 565
1
vote
0
answers
337
views
What is the standard definition of dual of disconnected planar graph when underlying graph derives 'product structure' over connected graphs?
Dual graph of a plane graph has a standard definition https://en.wikipedia.org/wiki/Dual_graph and an edgeless graph on $n$ vertices is planar. What is the standard dual graph of such a graph?
Update ...
- 1,826
4
votes
0
answers
281
views
Reference for results about planar graphs
A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
- 2,339
4
votes
0
answers
108
views
Reference on generalization of plane graph duality between bonds and simple cycles
Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a ...
- 1,462