5
votes
4answers
233 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
1answer
66 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 ...
5
votes
2answers
201 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
1answer
98 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 ...
2
votes
4answers
2k views

Graduate Schools for Graph Theory

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
2answers
458 views

spanning trees of plane graphs

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
0answers
145 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
1answer
176 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
1answer
313 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
1answer
364 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 ...
12
votes
3answers
460 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 ...
13
votes
2answers
673 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 ...
18
votes
3answers
735 views

Obstructions for planar graphs on surfaces of genus g

Kurotowski'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 ...
11
votes
1answer
446 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
2answers
869 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 ...
4
votes
2answers
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
2answers
3k 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
0answers
710 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, ...
3
votes
3answers
757 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 ...
3
votes
1answer
1k 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
2answers
466 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 ...
28
votes
3answers
4k 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, ...