Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [coloring]

For questions concerned with graph colorings.

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring.

Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

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Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.

Problem. Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational. I have tried to prove it in the following way, but I am not…
Arpon Basu
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Four color theorem disproof?

My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it. After some time scribbling…
Doktor J
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Is Wolfram wrong about unique 3-colorability, or am I just confused?

The illustration on Wolfram's page claims to present a uniquely colorable, triangle-free graph. However, this seems to be blatantly false: the graph has a symmetry with respect to a reflection through the horizontal axis, and we can use this…
Jakub Konieczny
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How to colour the US map with Yellow, Green, Red and Blue to minimize the number of states with the color of Green

I want to colour the US (only the states) map with Yellow, Green, Red and Blue. I was wondering what would be the lowest number of states with the colour of Green. We can of course use the other colours as much as we want. Please note that I want to…
Sina Babaei Zadeh
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Every point of a grid is colored in blue, red or green. How to prove there is a monochromatic rectangle?

I have a $3$-coloring of $\mathbb{Z}\times\mathbb{Z}$, i.e. a function $f:\mathbb{Z}\times\mathbb{Z}\to\{\color{red}{\text{red}},\color{green}{\text{green}},\color{blue}{\text{blue}}\}$. I have to prove that there is a monochromatic rectangle with…
user366454
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Monochromatic squares in a colored plane

Color every point in the real plane using the colors blue,yellow only. It can be shown that there exists a rectangle that has all vertices with the same color. Is it possible to show that there exists a square that has all vertices with the same…
Amr
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Did the Appel/Haken graph colouring (four colour map) proof really not contribute to understanding?

I hope this isn't off topic - sorry if I'm wrong. In 1976, Kenneth Appel and Wolfgang Haken proved the claim (conjecture) that a map can always be coloured with four colours, with no adjacent regions given the same colour. This was controversial…
user510
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4 Color Theorem - What am I not seeing??

Let me say first that I am in no way a mathematician. Just slightly interested in mathematics. I think I may have found an exception to the 4 color theorem. I don't claim to be smarter than those who proved the theorem, and I'll assume I'm…
Tyler_Hutchins
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We call a coloring of $3$-regular graph with $3$ colors good if for every $3$ edges incident with a vertex ...

Let $G$ be a $3$-regular graph with $n$ vertices. Color each edge with red, blue or yellow. Now, we call a coloring of graph as good if any three edges incident with any vertex have one color or three colors. Prove that the number of good coloring…
nonuser
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How many 2-edge-colourings of $K_n$ are there?

I'm writing a paper on Ramsey Theory and it would be interesting and useful to know the number of essentially different 2-edge-colourings of $K_n$ there are. By that I mean the number of essentially different maps $\chi:E(K_n)\to\{1,2\}$. Of course,…
Clum
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Edge-coloring of bipartite graphs

A theorem of König says that Any bipartite graph $G$ has an edge-coloring with $\Delta(G)$ (maximal degree) colors. This document proves it on page 4 by: Proving the theorem for regular bipartite graphs; Claiming that if $G$ bipartite, but not…
Klaus
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"Math Lotto" Tickets - finding the minimum winning set

"Math lotto" is played as follows: a player marks six squares on a 6x6 square. Then six "losing squares" are drawn. A player wins if none of the losing squares are marked on his lottery ticket. 1)Prove that one can complete nine lottery tickets…
Asher2211
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What made the proof of the four color theorem on planes so hard?

What made the proof of the four color theorem for planar graphs so hard? Analogous theorems on different objects (e.g. the torus) were proven long before the planar (spherical) case. Why was the planar case so hard? Or considering that "why" isn't a…
peterh
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Is it possible to uniquely number faces of a hexagonal grid with consecutive numbers?

You have a grid of regular hexagons. The aim of the game is to have each hex contain the numbers 1-6 on its edges. Each edge must also be connected to another edge that has a value one higher and one lower than the value of itself, with 6's wrapping…
Jonathan Van Buren
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Edge coloring of the cube

We have a cube and we are coloring its edges. There are three colors available. We say that the two colorings are the same if one can obtain a second by turning cube and permuting colors. Find the number of different colorings. Any ideas? I've found…
xan
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