04 - Connections - Relier d'un trait - 04

1 Points and Lines

Königsberg, 1736: is it possible to go through the city by crossing each of its seven bridges once and only once?
To resolve this problem, Euler holds the essential information: the city is divided into four districts represented by four points, connected by seven lines which symbolise the seven bridges.
The problem is then: on this plan, is there a road which passes only once over each line? It is the beginning of graph theory. Euler's answer: how many points are there where an odd number of lines ends.
There is a solution if this number is equal to zero or two!
• Leonhard Euler (1707-1783)

2 Are 4 colors enough?

How many colours are enough to colour a map in such a way that two adjacent countries are of different colours?

Graph theory allows us to model this problem and to reduce the number of cases to be studied to a finite (but still large) number. But thanks to the computer all these cases have been analysed.

Graph theory is used to model and study very important situations like telecommunication networks, electronic circuits, distribution networks -water, gas, electricity, post...- and numerous problems of logistics, transport and production.

3 Hello! Is that you?

How is your phone call connected?
It travels from relay to relay right up to the station nearest to your correspondent who is alerted by a bell.

In a town, these stations of the telephone network are best placed by taking into account the irregular topology of the streets.
Each station defines a zone of proximity of the call in the shape of a polygon which is connected to its neighbours.

These zones define a pavement of the city, called Voronoď's mosaic. If one connects the stations in neighbouring areas, one obtains a randomly determined graph which represents the cables along which the call will travel.
Graphs, probability theory and geometry all join to allow you to communicate!

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