The graph-colorings tag has no usage guidance.

**25**

votes

**2**answers

785 views

### chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following:
If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required?
This ...

**11**

votes

**0**answers

1k views

### A new lower bound for the chromatic number of a graph?

Let $S_{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph $G$. Let $S_{-}(G)$ denote the sum of the squares of the negative eigenvalues and $q$ the ...

**2**

votes

**0**answers

185 views

### Maximum number of 4-colorings of planar graphs (precise version)

This is a redo of my earlier question.
I'm trying again with more precision. I ignored one of the dicta in the "how to ask" page.
For a fixed $n$, what is known (references preferred) about the ...

**3**

votes

**3**answers

366 views

### Maximum number of different 4-colorings of planar graphs of a given size

I understand that it is computationally hard to count the 4-colorings of a given graph. See answers to
this question
In a given class of planar graphs (will leave choice of class open) is there a ...

**3**

votes

**1**answer

567 views

### Graphical representation of chromatic polynomial

Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that
$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle.
Also, we know ...

**0**

votes

**1**answer

106 views

### Counting walks on proper colorings of odd cycles

Let $G$ be an undirected odd cycle. Let $f$ be a proper 3-coloring of $G$. If $w=v_1v_2...v_k$ is a walk on $k$ vertices of $G$, let $f(w)=f(v_1)f(v_2)...f(v_k)$. Let $W_k=\{f(w)|w$ is a walk on ...

**7**

votes

**1**answer

1k views

### Has anyone seen this graph?

I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone ...

**6**

votes

**0**answers

342 views

### What is this subclass of $k$-colorable graphs called?

The following property emerged naturally when I was playing with certain generalizations of Kneser graphs.
Let $k>0$ be a natural number. Consider a property $P_k$ of graphs defined as follows.
...

**3**

votes

**3**answers

331 views

### Can you prove that hypergraphs with n-1 edges are partially 2 colorable?

I can. But my proof uses a theorem (which I do not reveal yet to avoid influencing you) and it feels like an overkill, so I wonder if there is a simple proof. Now the problem.
Suppose we have a ...

**0**

votes

**0**answers

199 views

### r-chromatic graphs with sufficient girth

Hi. For k,l positive integers let h(k,l) be the least integer with the property that in graph on h(k,l) vertices either there is a closed circuit of k or fewer lines or that the graph contains l ...

**1**

vote

**1**answer

825 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, ...

**3**

votes

**5**answers

1k views

### How many “different” colorings (excluding exchanges) exist for a given map (graph)?

In particular I'm interested in regular maps, excluding all maps that can be colored with 2 or 3 colors.
For what I need to analyze, maps have to be regarded as differently colored, if the same ...

**26**

votes

**4**answers

2k views

### Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples:
1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...

**1**

vote

**2**answers

1k views

### code that produces all possible trees with n nodes. [closed]

I'm looking for code that produces all possible trees with no self edges (or their adjacent matrices) with n nodes, anyone have any idea if this is written anywhere?

**4**

votes

**1**answer

944 views

### Simplest examples of unique-solution and unsolvable-without-backtracking Sudoku-like problems

A
The Sudoku game admits a broad generalization as follows : let $r$ be an integer $\geq 2$
and let $X$ be a finite set, and ${\cal X}$ be a collection of $r$-subsets of $X$
(i.e, a $r$-uniform ...

**4**

votes

**1**answer

264 views

### Is the chromatic number of the real plane invariant under the norm?

Recall that chromatic number of $\mathbb{R}^2$ is the least $n$ such that there exists a function $f$ from $\mathbb{R}^2$ into a set of colors ${C_1,\ldots,C_n}$ with $f(x)\neq f(y)$ for ...

**3**

votes

**2**answers

454 views

### Edge-coloring of a graph - special case

Hi everyone,
I have a problem I am working on that can be reduced to the following special case of edge coloring.
Let $G = (V,E)$ be an arbitrary graph. Furthermore, let each edge be assigned a ...

**5**

votes

**2**answers

529 views

### 4-coloring maps of pentagons

Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose
regions are pentagons? I am in fact most interested in convex pentagons, if that additional
structure ...

**4**

votes

**2**answers

685 views

### Coloring edges on a graph s.t. the set of edges for any two vertices have no more than 'k' colors in common

Please imagine the case where one has a planar graph, $G$, with a set of $|V|$ vertices, $(v_1, ..., v_{|V|}) \in V$, and $|E|$ edges, $(e_1, ..., e_{|E|}) \in E$. Now, provided a total of $N$ ...

**9**

votes

**2**answers

430 views

### Labelled spanning trees without edge crossings

Draw the complete graph $K_n$ on a plane in general position with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing ...

**4**

votes

**1**answer

668 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

134 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

539 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

448 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

666 views

### 4-colorable graphs

Hey guys, 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 iff each sub-graph G' in G ...

**4**

votes

**1**answer

421 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 ...

**30**

votes

**0**answers

939 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

480 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

1k 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

3k 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

413 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 ...

**28**

votes

**13**answers

4k 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

319 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

645 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

537 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

91 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 ...

**26**

votes

**3**answers

3k 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 ...

**3**

votes

**2**answers

626 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

632 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 ...

**17**

votes

**8**answers

4k 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

346 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

758 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 ...

**8**

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 ...