**2**

votes

**2**answers

175 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

89 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 ...

**3**

votes

**2**answers

159 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 ...

**1**

vote

**0**answers

67 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 ...

**2**

votes

**1**answer

109 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 ...

**3**

votes

**1**answer

122 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 ...

**7**

votes

**3**answers

247 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 ...

**7**

votes

**2**answers

442 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 ...

**4**

votes

**0**answers

177 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, ...

**9**

votes

**5**answers

393 views

### Do there exist sparse graphs with large crossing number?

Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do ...

**1**

vote

**1**answer

104 views

### Two definitions of genus for circle graphs

In the (very nice) article of Goldstein and Turner untitled Applications of Topological Graph Theory to Group Theory, the following definitions can be found:
Definitions: A circle graph is a pair $(G,...

**5**

votes

**2**answers

269 views

### Genus of Tutte-Coxeter Graph

What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...

**4**

votes

**1**answer

124 views

### genus of a finite simple undirected graph

Let $G$ be a finite simple undirected graph. Suppose there exist subgraphs $G_1,G_2,\dots,G_n$ of $G$, such that $G_i$ and $G_j$, have no common edges and have at most two common vertices, for each $...

**3**

votes

**0**answers

95 views

### Computing with Graphs in Surfaces

I asked this question yesterday on math.stackexchange, but the only response so far hasn't really addressed the question, so I thought I'd cross-post it.
I am currently working on a research project ...

**1**

vote

**5**answers

4k views

### Graduate Schools for Graph Theory [closed]

I am a rising senior in a small liberal arts college, and I was wondering if anyone could suggest me good graduate schools for graph theory. My only exposure to graph theory has been the intro graph ...

**8**

votes

**2**answers

663 views

### Spanning trees of plane graphs containing an edge of every face

I feel sure this must be known, but can I find it??
Which connected plane graphs (graphs drawn in the plane without crossings) have a spanning tree such that at least one edge of each face is in the ...

**2**

votes

**0**answers

163 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 ...

**2**

votes

**1**answer

197 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? ...

**2**

votes

**1**answer

333 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 ...

**9**

votes

**1**answer

378 views

### Flow on Infinite Graphs

Assume you have a simple, infinite graph $G$ with bounded degree (there is an upper bound for the degree of the nodes). Choose an arbitrary vertex $x\in V(G)$ and consider
$$
G_{n}:=\{x\in G:d(x_0,x)\...

**12**

votes

**3**answers

477 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 ...

**8**

votes

**2**answers

310 views

### Spectral techniques for genus of a graph

A generic question:
are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.

**14**

votes

**2**answers

821 views

### The Klein bottle and the Heawood Conjecture

Let $\Sigma_g$ be a surface of genus $g$. The Heawood Conjecture gives a closed formula in one variable, $\chi$ (the Euler characterstic of $\Sigma_g$), for the minimal number of of colors needed to ...

**1**

vote

**1**answer

258 views

### Why are graph imbeddings defined the way they are?

In my recent question I asked about a proof for the fact that the dual of a dual graph imbedding is equal to the original graph. Thinking about this a little more leads me to wonder why graph ...

**2**

votes

**0**answers

189 views

### Double duality for “geometrically defined” graph imbeddings

I am studying imbeddings of (connected, undirected, unweighted, multi-)graphs on oriented surfaces of arbitrary genus, and I am searching for a reference for the statement that the dual of the dual ...

**19**

votes

**3**answers

978 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 ...

**12**

votes

**1**answer

549 views

### Homotopy theory for spanning trees of a graph

I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of ...

**11**

votes

**2**answers

1k 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 ...

**5**

votes

**2**answers

2k views

### Singular homology of a graph.

By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ ...

**4**

votes

**2**answers

5k views

### Reporting all faces in a planar graph

Hi, I was looking to traverse a planar graph and report all the faces in the graph (vertices in either clockwise or counterclockwise order). I have build a random planar graph generator that creates a ...

**6**

votes

**0**answers

790 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, ...

**4**

votes

**3**answers

869 views

### Is there a bipartite analog of graph theory?

I would like to compile a list of questions about graphs that have a non-trivial analogs for bipartite graphs.
Let me give the following examples:
Cycle vs Even cycle. Most questions about cycles ...

**2**

votes

**1**answer

2k views

### ? A graph is four colorable if and only if it is planar.

? A graph is four colorable if and only if it is planar.
Is this true, I know that if a graph is planar it is four colorable, but is it true that if a graph is four colorable it must be a planar ...

**1**

vote

**2**answers

547 views

### How many dimensions I need to embed a graph? [duplicate]

Possible Duplicate:
What is the max number of points in R^3, interconnected by generic curves?
Given a set of points connected by edges lying on an euclidean plane,
I'd like to find which is the ...

**38**

votes

**4**answers

6k 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, ...