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49
votes
9answers
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 ...
34
votes
9answers
3k 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 and strengthenings of the four color theorem ...
34
votes
6answers
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 ...
30
votes
0answers
940 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 ...
28
votes
13answers
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 ...
26
votes
4answers
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 ...
26
votes
3answers
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 ...
25
votes
2answers
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 ...
25
votes
0answers
633 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 = ...
23
votes
1answer
1k 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
3answers
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
1answer
919 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 ...
17
votes
8answers
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. ...
15
votes
1answer
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
3answers
754 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
6answers
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: $\chi(G) \geq \omega(G)$ ...
14
votes
0answers
195 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 ...
12
votes
6answers
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
3answers
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 ...
12
votes
1answer
310 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
1answer
336 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
0answers
229 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
votes
0answers
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
9answers
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
2answers
458 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
2answers
189 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
1answer
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 ...
10
votes
2answers
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
2answers
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 ...
9
votes
1answer
87 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
1answer
646 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
5answers
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 ...
8
votes
1answer
358 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
3answers
481 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
2answers
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 ...
8
votes
1answer
365 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 ...
7
votes
2answers
347 views

Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$). For odd integers $n$ we have $n=\chi(K_n) = ...
7
votes
2answers
240 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
2answers
154 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 ...
7
votes
2answers
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 ...
7
votes
1answer
261 views

A variant to the Hadwiger-Nelson problem

Consider the following graph $G=(V,E)$ where $V=\mathbb{R}^2$ and $E = \{\{x,y\}: x,y \in \mathbb{R}^2 \text{ and } |x-y|\in \mathbb{Q}\}$. What is $\chi(G)$? (This is a variant of the ...
7
votes
1answer
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
1answer
449 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
0answers
147 views

Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements: (H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$. ...
7
votes
0answers
206 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
votes
0answers
240 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 ...
6
votes
5answers
1k views

Highly symmetric 6-regular graph with 20 vertices

I'm interested in (node/edge-)symmetric 6-regular graphs on 20 vertices and 60 edges, especially ones with a A5/icosahedral/dodecahedral symmetry group and especially their chromatic number. So far I ...
6
votes
2answers
417 views

Minimal graphs of prescribed girth and chromatic number

The well known result of Erdős, states that Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$ What I am wondering is When can we ...
6
votes
1answer
284 views

coincidence between minimal triangulation numbers and chromatic numbers

A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 ...
6
votes
1answer
288 views

What is the status of this strong form of Hedetniemi's conjecture?

In this question, a graph is a finite, undirected graph without loops or multiple edges, and a colouring of a graph is a proper vertex colouring. The product $G \times H$ of graphs $G$ and $H$ is the ...