The graph-colorings tag has no usage guidance.

**49**

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**9**answers

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### 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 ...

**41**

votes

**12**answers

4k views

### Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors.
The purpose of this question is to collect generalizations, variations, and strengthenings of the four ...

**34**

votes

**6**answers

3k views

### Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...

**33**

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 ...

**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 ...

**31**

votes

**0**answers

977 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 ...

**30**

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 ...

**27**

votes

**0**answers

666 views

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**25**

votes

**2**answers

803 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 ...

**24**

votes

**1**answer

2k views

### Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946

I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph ...

**20**

votes

**3**answers

1k views

### Can one measure the infeasibility of four color proofs?

Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is ...

**19**

votes

**1**answer

952 views

### A Question on 1, 2 ,3 Conjecture

The 1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the ...

**18**

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.
...

**16**

votes

**0**answers

250 views

### Simpler proofs of certain Ramsey numbers

The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture.
But for bigger Ramsey ...

**16**

votes

**0**answers

244 views

### Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed.
Say that a coloring of the dots of a Ferrers ...

**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 ...

**14**

votes

**3**answers

788 views

### Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has 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: $\chi(G) \geq \omega(G)$ ...

**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 ...

**12**

votes

**3**answers

767 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 ...

**12**

votes

**1**answer

320 views

### Coloring the edges of a torus graph

Question:Consider the $2k \times 2k$ grid graph on a torus. Is it true that for every $2$-coloring of the edges, there is an antipodal pair of vertices connected by a path that changes colors at most ...

**11**

votes

**1**answer

215 views

### What is the chromatic number of the “conic hypergraph” on a non-singular plane cubic?

Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 distinct points of the same color?
If so, what is the smallest ...

**11**

votes

**1**answer

338 views

### Coloring $K_n$ via edge-weight sums

This is a question inspired by
and tangential to "A Question on 1, 2 ,3 Conjecture"—and certainly
much easier!
Suppose one assigns a random edge weight among $\{1,2,3,\ldots,k\}$ to each
edge ...

**11**

votes

**1**answer

135 views

### Graphs with a coloring that majorizes all other colorings

By a coloring of a graph $G = (V,E)$ I mean a map $\kappa:V\to\mathbb{N}$ such that $\kappa(u)\ne \kappa(v)$ whenever $u$ and $v$ are adjacent. (Sometimes this is called a proper coloring but I am ...

**11**

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**0**answers

256 views

### The Universal Labeling of graph

The universal labeling of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ ...

**11**

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**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 ...

**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 ...

**10**

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 ...

**10**

votes

**2**answers

484 views

### What is the smallest 4-chromatic graph of girth 5?

It is known that the smallest 4-chromatic graph of girth 4 is the Grötzsch graph (11 vertices). What happens for girth 5?
The Brinkmann graph (21 vertices) has chromatic number 4, girth 5 and is ...

**10**

votes

**2**answers

191 views

### Big Mono-Chromatic Subgraphs of Vertex 2-Colourings

I'm not a graph theorist, but the following quantity came up in my work and I'm curious if it has been studied. Given a connected finite graph $\Gamma = (V,E)$ define: $$ c(\Gamma) = \min_{f : V ...

**10**

votes

**1**answer

636 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 ...

**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 ...

**9**

votes

**2**answers

431 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 ...

**9**

votes

**1**answer

96 views

### A variant of Nelson-Hadwiger Problem on the chromatic number of the plane

The famous Nelson-Hadwiger problem asks about the chromatic number of the graph $G$, with the vertex set $V(G)={\mathbb R}^2$ where $a_1=(x_1,y_1), a_2=(x_2,y_2) \in V(G) \ $ form an edge iff ...

**9**

votes

**1**answer

650 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 ...

**8**

votes

**2**answers

182 views

### Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the
plane so that each arc is crossed at most once by another arc.
A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times.
Planar graphs are ...

**8**

votes

**1**answer

370 views

### Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...

**8**

votes

**3**answers

499 views

### Edge-coloring of the complete graph without any rainbow paths

For a given $2k-1$ edge coloring of the complete graph $K_{2k}$,
say a Hamiltonian path $P$ is a rainbow path if every color appears exactly once in $P$.
My question is
"For each $2k (k \geq 2)$, is ...

**8**

votes

**2**answers

546 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 ...

**8**

votes

**1**answer

369 views

### Postnikov's approach to perfect matchings of graphs

Over a decade ago Alexander Postnikov developed his own way of looking at perfect matchings of bipartite plane graphs. As I recall, he starts with a 2-coloring of the square grid and creates a new ...

**8**

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**0**answers

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### Edge-colorings of plane graphs: do you know references where the following questions are studied?

Let $G$ be a plane graph (or more generally, a graph embedded on a surface) with a proper edge-coloring of $G$ with $k$ colors $\{1,\ldots,k\}$. I am interested in studying the cyclic permutations of ...

**7**

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**4**answers

249 views

### Graph homomorphisms and line graph

If $G,H$ are simple, undirected graphs, we say that they are in a hom(omorphism)-relation if there is a graph homomorphism from $G$ to $H$ or from $H$ to $G$.
For any graph $G$ let $L(G)$ denote its ...

**7**

votes

**2**answers

247 views

### Conjecture: for perfect graphs the fractional chromatic index rounded up equals the chromatic index

Let $\chi'_f(G)$ be the fractional chromatic index.
Based on limited experiments (up to 8 vertices and few larger graphs),
I suspect:
Conjecture For perfect graphs $\lceil \chi'_f(G) \rceil = ...

**7**

votes

**2**answers

415 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 ...

**7**

votes

**1**answer

264 views

### Chromatic numbers of nowhere dense graphs

Given $c\in (0,1)$ and a graph $G=(V,E)$ such that any subset $U\subset V$ contains an independent subset of cardinality at least $c|U|$. Does it allow to bound the chromatic number $\chi(G)$ by the ...

**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 ...

**7**

votes

**1**answer

450 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 ...

**7**

votes

**1**answer

506 views

### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

**7**

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**0**answers

239 views

### Colouring a graph whose edge set is a special union of cliques

I am trying to show that a certain family of graphs can always be properly coloured with at most $6$ colours (where "properly coloured" means that each vertex gets a colour and no edge has both ends ...

**7**

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**0**answers

243 views

### A maximum discrepancy hypergraph 2-colouring problem

This is sort of a hypergraph-ish question that I feel should be easy to prove or disprove but I can't see it right now.
The setup is as follows. We have a vertex set partitioned in to sets ...