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# Questions tagged [graph-theory]

Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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### Triangle-free Lemma

Theorem (Triangle-free Lemma). For all $\eta>0$ there exists $c > 0$ and $n_0$ so that every graph $G$ on $n>n_0$ vertices, which contains at most $cn^3$ triangles can be made triangle free ...
1k views

### Ramsey Theory, monochromatic subgraphs

If we have the complete countably infinite bipartite graph $K_{\omega,\omega}$ and we colour the edges with just two colours. Should we expect to get a monochromatic copy of $K_{\omega,\omega}$. ...
1k views

### Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)

If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler'...
2k views

### “The” random tree

One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...
2k views

### Number theoretic spectral properties of random graphs

If G is a graph then its adjacency matrix has a distinguished Peron-Frobenius eigenvalue x. Consider the field Q(x). I'd like a result that says that if G is a "random graph" then the Galois group ...
6k views

### Graphs with fractal properties?

For the purposes of a research project, I am wondering if there are any resources on graphs with fractal properties, by which I mean self-similarity in particular. For instance, imagine a graph where ...
6k views

### Which graphs have incidence matrices of full rank?

This is a follow-up to a previous question. What graphs have incidence matrices of full rank? Obvious members of the class: complete graphs. Obvious counterexamples: Graph with more than two ...
870 views

### Is every matching of the hypercube graph extensible to a Hamiltonian cycle

Given that $Q_d$ is the hypercube graph of dimension $d$ then it is a known fact (not so trivial to prove though) that given a perfect matching $M$ of $Q_d$ ($d\geq 2$) it is possible to find another ...
581 views

### Algorithms for laying out directed graphs?

I have an acyclic digraph that I would like to draw in a pleasing way, but I am having trouble finding a suitable algorithm that fits my special case. My problem is that I want to fix the x-...
222 views

### Where to find nice diagrams of trees and other graphs? [closed]

Are there some publicly available, vector format diagrams of trees and other graphs? They aren't hard to make, but they sure do take a lot of time (for me).
420 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 ...
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### Graphs with incidence matrices whose pseudoinverses are proportional to their transposes

When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K_n$ (all nodes ...
20k views

### What introductory book on Graph Theory would you recommend?

I'm looking for a book with the description of basic types of graphs, terminology used in this field of Mathematics and main theorems. All in all, a good book to start with to be able to understand ...
1k views

### Erd?s–Stone theorem type edge density estimates for bipartite graphs?

The Erd?s–Stone theorem theory says that the densest graph not containing a graph H (which has chromatic number r) has number of edges equal to $(r-2)/(r-1) {n \choose 2}$ asymptotically. However, ...
956 views

### Number of paths equal less than equal to a certain length

Hey, I need to count the number of paths from node $s$ to $t$ in a weighted directed acyclic graph s.t. the total weight of each path is less than or equal to a certain weight $W$. I have an ...

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