**4**

votes

**1**answer

717 views

### Hypergraph Chromatic Number vs Degree, Clique-Size

For a hypergraph let $\chi$ be the least number of colours needed to colour the vertices, so that in each edge, each colour is used at most once (i.e., the strong chromatic number). Let $\Delta$ be ...

**2**

votes

**0**answers

135 views

### finding set of tree decompositions to cover all pairs of vertices

I first asked this on cstheory.SE but got no reply.
Let $P(X_i=x)$ represent probability that randomly chosen proper $q$-coloring of an $L\times L$ square grid contains color $x$ at position $i$. How ...

**8**

votes

**2**answers

552 views

### Derivative of Tutte polynomial at -1

Let Tutte polynomial on graph with edge-set $E$ be defined as follows
$$f(q,v)=\sum_{A\subseteq E} q^{\kappa(A)} v^{|A|}$$
Here the sum is over all subgraphs $A$, $\kappa(A)$ is the number of ...

**7**

votes

**1**answer

452 views

### Graphs with the same chromatic symmetric function

Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s ...

**2**

votes

**3**answers

757 views

### A conjectured criterion for 4-colorable graphs

I tried to find a solution to this in the web but couldn't.
Can you tell me if the following sentence is correct or else give me a counterexample?
$G$ is $4$-colorable if and only if each sub-graph ...

**4**

votes

**1**answer

431 views

### Bounds on strong vertex colourings of regular hypergraphs?

What are the best results for upper bounds on the number of colours required in a strong vertex colouring of a regular hypergraph H?
A regular hypergraph is one in which every vertex is contained in ...

**32**

votes

**0**answers

999 views

### Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...

**4**

votes

**2**answers

484 views

### Tractably Partitioning the possible vertex k-colorings of a graph by local stability and instability.

A k-coloring or k-labeling of the vertices of a single-component undirected graph G with $n$ vertices can be a proper coloring or not. If it is not a proper coloring, such that each vertex has ...

**-1**

votes

**1**answer

2k views

### Are all Hamiltonian planar graphs are 4 colorable? Does this imply all planar graphs are colorable?

Planar graphs with a Hamiltonian loop connecting all faces do not necessarily have a Hamiltonian on their edges, which would make a 3 edge coloring and thus a 4 face coloring easy. However they have a ...

**14**

votes

**6**answers

4k views

### Lower bounds for chromatic number of a graph

I am trying to find a good lower bound for chromatic number of one family of graphs. I'm curious what are the known lower bounds for chromatic number. There are two obvious: $\chi(G) \geq \omega(G)$ ...

**7**

votes

**2**answers

418 views

### Simple proof that these graphs are perfect

I recently came across a family of infinite graphs (in the context of two-dimensional convexity) that don't have induced 4-paths (paths with 4 vertices).
Note that the complement of a 4-path is again ...

**31**

votes

**15**answers

5k views

### Strengthening the Induction Hypothesis

Suppose you are trying to prove result $X$ by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result $X'$ (that you don't really care about) by induction. This ...

**49**

votes

**9**answers

6k views

### The “sensitivity” of 2-colorings of the d-dimensional integer lattice

Consider the $d$-dimensional integer lattice, $Z^d$. Call two points in $Z^d$ "neighbors" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate).
Let $C$ be a ...

**2**

votes

**2**answers

325 views

### Colourings of Graphs with extra conditions

As a phd-student I've wandered into a question of colourings of graphs and wondered what was known about them.
Given a finite graph G, where the maximum degree of a vertex is d, I'm interested in ...

**12**

votes

**6**answers

2k views

### A decision problem in graph coloring

It'll be great to get a pointer or answer to the following question:
What is the complexity of the following problem? Given an unweighted and undirected graph, can we have a proper (not necessarily ...

**9**

votes

**1**answer

659 views

### Finding Two Rainbow Spanning Trees

Suppose we have a graph whose edges are coloured. It's not necessarily a proper colouring: a given node may have 0, 1, or several incident edges of a given colour.
Is the following problem ...

**10**

votes

**2**answers

538 views

### Can we select a rainbow matching if each degree is 6 and each colorclass is a C_6?

Suppose that we have a 2d-regular graph whose edges are colored such that the edges of each color form a cycle of length 2d. (So if the graph has 2n vertices, then there are n colors.) Is it true that ...

**1**

vote

**1**answer

92 views

### Graphical representation of duals of n-simplices

Is there any way to graphically represents the duals of n-simplices?
For example, I want to know how exactly duals of tetrahedrons arrange between themselves, i.e how many faces, volumes share the ...

**34**

votes

**5**answers

4k views

### Algebraic Proof of 4-Colour Theorem?

4-Colour Theorem. Every planar graph is 4-colourable.
This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism ...

**15**

votes

**1**answer

1k views

### Making a non-monotone function monotone

Consider a function $f: \{0,1\}^n \rightarrow \{1..R\}$. This function can be interpreted as a coloring $Color(v)$ of vertices in a unit n-dimensional hypercube in $R$ colors.
We say there is a ...

**5**

votes

**4**answers

1k views

### Coloring Points in the Plane

Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...

**5**

votes

**2**answers

780 views

### Where can I find a catalog of known Ramsey numbers?

Is there an online catalog available of Ramsey numbers, preferably one that for unknown values documents the known upper/lower bounds?

**4**

votes

**7**answers

2k views

### Suggest effective heuristic (not precise) graph colouring algorithm

Can you suggest a good rough graph colouring algorithm? What are the best such algorithms nowadays?

**10**

votes

**1**answer

642 views

### Which lattices have more than one minimal periodic coloring?

The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of ...

**18**

votes

**8**answers

5k views

### Why is edge-coloring less interesting than vertex-coloring?

I was wondering why there is (apparently) much more research directed towards vertex-coloring than edge-coloring? Prima facie, it seems that edge-coloring is just as "natural" a thing to investigate.
...

**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 ...

**5**

votes

**2**answers

351 views

### How are graph automorphisms are affected by transformations?

I have a heavily symmetric regular graph whose automorphisms I know. I remove one subgraph and insert another one in a consistent manner; for example, this could be a Delta-Y transformation ...

**12**

votes

**3**answers

771 views

### Is there a matrix whose permanent counts 3-colorings?

Actually, I suppose that the answer is technically "yes," since computing the permanent is #P-complete, but that's not very satisfying. So here's what I mean:
Kirchhoff's theorem says that if you ...

**10**

votes

**9**answers

2k views

### What is the Tutte polynomial encoding?

Pretty much exactly what it says on the tin. Let G be a connected graph; then the Tutte polynomial T_G(x,y) carries a lot of information about G. However, it obviously doesn't encode everything about ...

**11**

votes

**5**answers

1k views

### Can one make Erdős's Ramsey lower bound explicit?

Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...