3
I.
Graph Theory
N6 15H
7
M4TH5 +
Euler in K¨ onigsberg
The Seven Bridges of K¨ onigsberg.
Map of K¨ onigsberg in the XVIIIth century K¨ onigsberg is a city which was the capital of East Prussia but now it is known as Kaliningrad in Russia. The city is built around the River Pregel where it joins another river. An island named Kniephof is in the middle of where the two rivers join. There were seven bridges that joined the different parts of the city on both sides of the rivers and the island. The ”Seven Bridges of K¨ onigsberg” is a notable historical problem in mathematics. Its resolution by Leonhard Euler (1707 (Bˆale, Switzerland) – 1783 (St Petersburg, Russia)) in 1735 laid the foundations of graph theory. The problem was stated by Euler while visiting the city: Find a walk through the city that crosses each bridge once and only once. ✄ .............................................................................................................. C g c
d e
A a
b
D
f
B
Map of K¨ onigsberg and its graph. Euler had the idea to represent each of the boroughs of the town as a vertex. The bridges (a, b, c, d, e, f, g) are lines called edges connecting the four vertices A, B, C and D. This drawing is now called a graph. The problem stated as ”Find a walk through the city that crosses each bridge once and only once.” would now be stated as ”Find a path going along each edge once and only once.” If such a path exists, we call it eulerian path. If there is a eulerian path that starts and ends at the same vertex, it is called a eulerian tour (or eulerian circuit) ✩ Look at the graphs below and search for a eulerian tour or a eulerian path. ✪ Calculate the degree of each vertex. The degree of a vertex is the number of edges connected to this vertex.
①
②
③
④
⑤
⑥
⑦
⑧
⑨
⑩
3
II.
N6 15H
7
M4TH5 +
Graph Theory
Eulerian graphs
Eulerian graphs
II..1
The theorems
Euler found and proved the two following theorems that give the necessary and sufficient conditions under which eulerian tours and paths exist.
Euler’s theorems. Theorem 1. A graph G possesses an eulerian tour iff each vertex of G has an even degree. Theorem 2. A graph G possesses an eulerian path iff G contains exactly zero or two vertices of odd degree. Moreover if there are two vertices of odd degree, these must be the starting and ending points of any eulerian path on G ✫ Check these two theorems on the ten graphs we studied.
II..2
Fishy business
The resort was far too charming a place for such a senseless, smelly crime. Seven rustic bungalows, some of them on a lagoon (A through D) and some on the oceanfront (F and G), were connected by pathways as shown in the figure. A fisherman saw a sinister-looking man approach the resort from the lagoon carrying a large basket and sneak into one of the bungalows bordering that body of water. This man then stalked along the pathways from one bungalow to the next, leaving rotting fish everywhere. Police detectives determined from a set of muddy footprints that the madman had traveled along each pathway exactly once. The detectives saw no footprints leading away from the resort, so they concluded that the fish vandal was still hiding in one of the bungalows. Unfortunately, the footprints on the pathways were so indistinct that the detectives couldn’t tell the direction in which they pointed. What is more, the fisherman couldn’t remember which of the four bungalows on the lagoon the vandal had first entered. The police were therefore unable to retrace the deviant’s route but what they knew was that he’d never walked the same pathway twice.
① Can you find the one bungalow in which the fish vandal must be hiding? ② What if the fish vandal could retrace at most one path but must still cover all paths and still come in on the lagoon side. Then where might he be?
N6 15H
Graph Theory Chinese postman problem
M4TH5 +
3
III.
The Chinese postman problem
7
In graph theory there are two important problems that both arise when traveling inside a city or from a city to another city: The travelling salesman problem asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? The Chinese postman problem is asking for the shortest tour of a graph which visits each edge at least once. This problem was originally studied by the Chinese mathematician Mei-Ku Kuan in 1962. 1. Explain the names given to these problems. Are they different problems? 2. First problem. On his first day on the job, a new postman is given the map below and instructed that he must deliver letters to houses along each of the streets on the map. The postman wants to find the shortest possible route that includes each of the streets on the map at least once and returns him to his starting point at the post office (the intersection in the upper lefthand corner).
3. Try to solve the same problem with the three following maps. The post office is point A.
N6 15H
7
3
M4TH5 +
Graph Theory Chinese postman problem
Oral questions:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Warm up: ① Can you see the seven bridges on the map above? ② Playschool: Color the river with blue and the bridges with another color. ③ Each bridge allows to go from one borough
to another. How many boroughs of this town can you see? ④ Could you draw a simpler map of the city just with the river and the bridges? ⑤ Try to draw a solution with the help of your neighbour.
............................................................................................. ① Some of the graphs above are ”the same”. Can you find them? Answer: 3& 9; 4&7 ② What does ”are the same” mean here for a graph. ③ When you found a path, could you find another one? ④ Can you start at any vertex? Same question for a tour. ⑤ Look for necessary conditions in order that a graph has a eulerian tour or path. ⑥ Hint: we call degree of a vertex the number of edges connected to this vertex. ⑦ If you want a eulerian tour starting at vertex A, what must be the degree of A? ⑧ If you have a eulerian tour starting at vertex A,can you start from another vertex?
