All Questions
Tagged with topological-graph-theory co.combinatorics
35 questions
47
votes
4
answers
10k
views
Why are planar graphs so exceptional?
As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, ...
- 12.6k
25
votes
1
answer
596
views
Doubly periodic 4 color theorem?
Let $G$ be a graph embedded (without crossings) on a torus $T$. It's fairly well known that this implies the chromatic number of $G$ is at most 7. If I lift $G$ to the universal cover of $T$, we get a ...
- 2,242
24
votes
3
answers
2k
views
Gauss-Bonnet Theorem for Graphs?
One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an $n$-cycle has $\chi = 0$ and $K_4$ has $\chi=-2$.
Is there an analog for the ...
- 151k
21
votes
3
answers
2k
views
Obstructions for embedding a graph on a surface of genus g
Kuratowski's theorem tells us the complete graph $K_5$ and the bipartite graph $K_{3,3}$ are the only obstructions to a graph being planar, ie embeddable in the plane with no edge-crossings.
Is the ...
- 1,180
12
votes
3
answers
582
views
Can we map every graph in the plane such that all induced cycles selfintersect?
Suppose we have a graph G. Is it true that we can map its vertices to the plane such that when connecting neighboring vertices with segments, then any induced cycle of G that has length at least 4 ...
- 18.7k
11
votes
2
answers
2k
views
Given a graph embedded on a torus, how many edges are necessary for noncontractible loops to be long?
If we are given a graph embedded on a torus, with the following properties, what is the minimum number of edges it can have?
Any noncontractible loop is comprised of at least n edges.
Any ...
- 111
10
votes
1
answer
369
views
Orientations of Planar Graphs
Let $G$ be a $2$-edge-connected graph drawn in the plane (such that the edges intersect only at the endpoints). I want to orient
the edges of $G$ such that for each vertex $v$, there are no
three ...
- 175
10
votes
1
answer
312
views
Is this drawing of $K_{4,4}$ knotted?
Let $A$ and $B$ be skew lines in $\mathbb{R}^3$. Choose four points $a_1, a_2, a_3, a_4$ on $A$ and four points $b_1, b_2, b_3, b_4$. For all $i,j \in [4]$ draw a line segment from $a_i$ to $b_j$. ...
- 32.1k
9
votes
3
answers
2k
views
Embedding planar graphs into the grid
I've seen the following lemma in a paper. The result is by Valiant.
A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
- 903
9
votes
0
answers
1k
views
Simplicial Representations of (Hyper)Graph Complexes
For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, ...
- 1,324
8
votes
2
answers
615
views
Embedding of planar graphs
I've recently come across the following lemma.
Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
- 83
6
votes
1
answer
142
views
Embedding linklessly embeddable graphs without Borromean rings
A linklessly embeddable graph is a graph which can be embedded into $\Bbb R^3$ so that no two of its cycles are linked. For example, the Petersen graph is not such a graph.
Now, I can think of another ...
- 13.6k
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
5
votes
0
answers
83
views
When does the ΔY-family of a simple graph contain multigraphs?
Given a graph $G$, its ΔY-family is the smallest family of graphs that contains $G$ and is closed under ΔY- and YΔ-transformations.
Since YΔ-transformations can introduce multi-edges, the ΔY-family of ...
- 13.6k
4
votes
2
answers
232
views
Number of non-equivalent graph embeddings
Given a graph $G$, there is a minimal integer $g$ associated with it which captures the minimum genus a surface needs to have so that $G$ embeds in the surface without edge crossings.
Is there a way ...
- 13.9k
4
votes
1
answer
214
views
Find all 2-planar drawings of $K_6$ and $K_7$
A $k$-planar graph is a graph which can be embedded with at most $k$ crossings per
edge.
It is proved that a complete graph $K_n$ is 2-planar if and only if $n\le 7$.
Angelini P., Bekos M. A., ...
- 1,851
4
votes
2
answers
266
views
Asymptotics of list size in Robertson-Seymour theorem
A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are ...
user76479
4
votes
1
answer
540
views
Connection between connectivity and cohesion of a graph
Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...
user39815
3
votes
1
answer
163
views
Conditions on graphs to assure unique embedding on a fixed genus surface
The classical Whitney Theorem in low topological theory/graph theory states that every 3-connected planar graph is uniquely embeddable (up to orientation) on the sphere.
My question is the following: ...
- 1,561
3
votes
1
answer
215
views
Construction of planar embedding
I'm reading the following paper on universality considerations in VLSI circuits
http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
In Theorem 2 On the second page it states there exists ...
- 903
3
votes
0
answers
94
views
The pagenumber of subdivision of a complete graph
A book embedding of a graph $G$ consists of placing the vertices of $G$ on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The book thickness $...
- 767
2
votes
1
answer
319
views
Why are graph embeddings defined the way they are?
In my recent question I asked about a proof for the fact that the dual of a dual graph embedding is equal to the original graph. Thinking about this a little more leads me to wonder why graph ...
- 2,830
2
votes
2
answers
413
views
Decomposing a graph into n-cycles [closed]
Suppose I have a strongly $k-regular$ graph $G$, of size $v$, where every vertex is $N>0$ $n-cycles$, for $at least$ one value of $n$ that divides $v$. Can we cut edges from $G$ in such a way ...
2
votes
1
answer
121
views
Orthogonal embeddings and edge lengths
I'm interested in orthogonal embeddings of graphs into the 2-dimensional, i.e where vertices are placed at integer co-ordinates and edges are routed along the grid lines and are not allowed to ...
- 903
2
votes
1
answer
137
views
VLSI circuit embeddings
In the following paper by Valiant
http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
He shows under theorem 2 (at the bottom of the second page) that any planar graph $G$ of degree 3 or 4 ...
- 903
2
votes
1
answer
238
views
Maximum fixed genus Bipartite graphs
Say $B_{n,n}$ is a bipartite graph on $2n$ vertices with $n$ vertices of color $1$ and with $n$ vertices of color $2$.
What is the maximum number of edges that a genus $g$ graph $B_{n,n}$ can have? ...
- 13.9k
2
votes
1
answer
401
views
Obstructions to genus $g+1$ bipartite graph having genus $g$
Say $B_{n,n}$ is a bipartite graph on $2n$ vertices with each color assigned to $n$ vertices.
Say I know $g \le \operatorname{genus}(B_{n,n}) \le g+1$. What obstructions prevent $B_{n,n}$ from being ...
2
votes
1
answer
111
views
Maximum genus of an abstract "cycle complex"
Let us define an abstract "cycle complex" as the following combinatorial object: it is $(V, C)$, where $V$ is a set of $n$ nodes, $C$ is a set of $c$ cyclically ordered subsets of $V$, each ...
- 1,389
2
votes
0
answers
215
views
Polyhedral embeddings of large face-width where all faces have the same length
Where can I find examples of polyhedral embeddings of simple graph with large face-width, such that all the faces have the same length?
By polyhedral embedding I mean an embedding of the graph on a ...
- 884
1
vote
1
answer
194
views
Bounds on lengths of intervals in bounded-degree interval graphs
A graph is said to be an interval graph if its vertices can be associated with (closed) intervals on the real line $\mathbb R$ and there is an edge between two vertices if and only if the ...
- 277
1
vote
1
answer
115
views
Bounds on lengths of boxes in bounded-degree box graphs
$\DeclareMathOperator{\box}{\operatorname{box}}$$\DeclareMathOperator{\cub}{\operatorname{cub}}$
This is a follow up and an extension of another question I asked recently.
A box graph is a graph ...
- 277
1
vote
0
answers
42
views
What lower bounds are known for pair crossing number and related questions in multigraphs?
So in terms of crossing number https://arxiv.org/pdf/1808.10480 gives a lower bound of $O(e^{2.5}/n^{1.5})$ for multigraphs with no face of length 2 with no node contained inside.
What do we know ...
- 111
1
vote
0
answers
33
views
Arranging bounded degree graphs into grids with few edges connecting horizontal and vertical lines
The following question arose when I was trying to find explicit topological embeddings of bounded degree graphs into $\mathbb R^3$ which match (asymptotically) the minimal possible "volume" ...
- 743
1
vote
0
answers
123
views
Building an orthogonal embedding for a 4-planar graph
I'm interested in the following paper http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
In particular i'm interested in the construction Valiant describes to prove that it is possible to ...
- 903
0
votes
0
answers
56
views
Are total graph of power of cycles homeomorphic to powers of cycles?
Is the total graph associated to powers of cycles homeomorphic to powers of cycles themselves?
I think yes, because the total graph associated to cycles is homeomorphic to cycles(i think?)So, does ...
- 2,089