# Tagged Questions

**1**

vote

**2**answers

122 views

### TSP, but for all routes not all points

Hello
I am trying to solve a problem involving a cross-country ski trail map. I wish to travel every trail on the map, at least once, but no more than twice (so I can out-and-back on a destination, ...

**8**

votes

**2**answers

338 views

### Is there an analog of 'Directed Graph' for topological spaces?

In a directed graph, certain paths are distinguished: Those that follow the edge direction on each edge. I was wondering if one could (or does) formulate a notion of a 'Directed Topological Space', on ...

**7**

votes

**2**answers

235 views

### Higher-dimensional Fáry's theorem?

Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges.
For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex ...

**26**

votes

**1**answer

1k views

### Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently ...

**2**

votes

**0**answers

86 views

### Graphs, non-Hausdorfness and Wallman compactifications of non-regular spaces

Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) ...

**3**

votes

**1**answer

386 views

### Klein Bottle exception to the Heawood Conjecture [duplicate]

Possible Duplicate:
The Klein bottle and the Heawood Conjecture
It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a ...

**36**

votes

**8**answers

4k views

### What is a continuous path?

I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're getting nervous ...

**0**

votes

**1**answer

230 views

### “Skein” equations sets that can reduce any graph

Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...

**2**

votes

**2**answers

995 views

### How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED

For referencing, I keep the original title and post and ask
only about the simplest case (and forget the freeway with crossings
for now).
Consider a trivalent graph, e.g. the dodecahedron or cube ...

**1**

vote

**1**answer

715 views

### Representations of regular maps (four color theorem)

For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem.
For example, ...

**4**

votes

**2**answers

458 views

### Is there a formula to count how many different topological regular maps can be created with n faces (on a sphere)?

Notes:
For "regular" I intend maps in which the boundaries form a 3-regular planar graph
For "different" I intend maps that cannot be topologically transformed one into another (faces ...

**1**

vote

**1**answer

256 views

### Winding number bijection on graphs

Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...

**6**

votes

**6**answers

1k views

### Elegant representations of graphs in R^3

If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any ...

**13**

votes

**2**answers

647 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 ...

**11**

votes

**1**answer

437 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 ...

**8**

votes

**2**answers

1k views

### How many simply connected subsets of an n-by-m grid?

Given an n-by-m square grid graph, how many ways are there to choose a subset of the vertices which is simply connected? Here, a subset of vertices is simply connected if the vertices, together with ...

**9**

votes

**2**answers

632 views

### Is there a way to see a topological group as the “Cayley graph” of its “infinitesimal generators”?

At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...

**1**

vote

**2**answers

1k views

### Maximum number of shortest-paths

I would like to know if there is a equation for the maximum number of shortest paths that pass through r where r is a node contained in any path from node s (a fixed node, i mean, s is the only source ...

**-6**

votes

**4**answers

555 views

### What is the max number of points in R^3, interconnected by generic curves?

The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of ...

**2**

votes

**4**answers

9k views

### Difference between connected vs strongly connected vs complete graphs [closed]

What is the difference between
connected
strongly-connected and
complete?
My understanding is:
connected: you can get to every vertex from every other vertex.
strongly connected: every vertex ...

**9**

votes

**3**answers

311 views

### Groupoid of moves on trivalent fatgraph

Let $T$ be a finite trivalent fatgraph - i.e. a graph with a cyclic order of the edges at each vertex. Then there are certain basic "moves" we can perform on $T$: an embedded edge can be collapsed and ...

**4**

votes

**5**answers

925 views

### A walk on a compact 2D surface embedded in 3-space that never returns home

At the risk of asking an uninformed question...
Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once ...