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1answer
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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
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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: $\chi(G) \geq \omega(G)$ ...
7
votes
2answers
410 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 ...
25
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12answers
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 ...
47
votes
8answers
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
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2answers
314 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
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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 ...
9
votes
1answer
621 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
2answers
533 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
1answer
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 ...
24
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 ...
15
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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 ...
5
votes
4answers
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
2answers
531 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
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7answers
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
1answer
612 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
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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. ...
2
votes
1answer
1k 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
2answers
340 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 ...
11
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3answers
735 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
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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 ...
8
votes
5answers
963 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 ...