Using Euler s Theorem

Transcription

1 Using Euler s Theorem Suppose that a connected, planar graph has 249 vertices and 57 faces. How many edges does it have? A: 106 B: 194 C: 304 D: 306 E: We don t have enough information

2 Using Euler s Theorem Suppose that a connected, planar graph has 249 vertices and 57 faces. How many edges does it have? A: 106 B: 194 C: 304 D: 306 E: We don t have enough information We have 2 = v e + f = 249 e + 57 = 306 e e = 304

3 Using Euler s Theorem Suppose that a connected, planar graph has 249 vertices and 57 faces. What is the sum of the degrees of the faces? A: 57 B: 114 C: 498 D: 608 E: We don t have enough information

4 Using Euler s Theorem Suppose that a connected, planar graph has 249 vertices and 57 faces. What is the sum of the degrees of the faces? A: 57 B: 114 C: 498 D: 608 E: We don t have enough information The sum of the degrees of the faces is twice the number of edges: sum of degrees of faces = = 608

5 Kuratowski s Theorem If G is a non-planar graph, then we can simplify the graph by deleting edges deleting vertices (and all adjacent edges) replace with

6 Kuratowski s Theorem If G is a non-planar graph, then we can simplify the graph by deleting edges deleting vertices (and all adjacent edges) replace with and end up with either: or K 5 K 3,3

7 Kuratowski s Theorem In other words, K 5 and K 3,3 are the simplest non-planar graphs K 5 K3,3

8 Kuratowski s Theorem In other words, K 5 and K 3,3 are the simplest non-planar graphs K 5 K3,3 We still need to see that K 5 and K 3,3 aren t planar

9 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? A: 5 B: 7 C: 12 D: None of the above

10 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? A: 5 B: 7 C: 12 D: None of the above v = 5 and e = 10

11 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? A: 5 B: 7 C: 12 D: None of the above v = 5 and e = 10 2 = v e + f = f f = 7

12 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? A: 10 B: 14 C: 20 D: None of the above

13 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? A: 10 B: 14 C: 20 D: None of the above The sum of the degrees of the faces is 2e = 20

14 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? 20 The average degree of a face is 20 7 < 3

15 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? 20 The average degree of a face is 20 7 < 3 So at least one face has to have degree 2

16 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? 20 The average degree of a face is 20 7 < 3 So at least one face has to have degree 2 Except for the graph every face of every graph has to have degree 3

17 K 5 is not Planar Consider K 5 : Suppose that K 5 is planar. How many faces will it have? 7 What is the sum of the degrees of the faces? 20 The average degree of a face is 20 7 < 3 So at least one face has to have degree 2 Except for the graph every face of every graph has to have degree 3 This is a contradiction, so K 5 is not planar.

18 Three Related Questions The following questions are related:

19 Three Related Questions The following questions are related: 1. Students are taking various classes with final exams. How many time slots do we need so that no student has a conflict?

20 Three Related Questions The following questions are related: 1. Students are taking various classes with final exams. How many time slots do we need so that no student has a conflict? 2. We have a map. What is the fewest colours we can use to colour it, so that adjacent countries are different colours?

21 Three Related Questions The following questions are related: 1. Students are taking various classes with final exams. How many time slots do we need so that no student has a conflict? 2. We have a map. What is the fewest colours we can use to colour it, so that adjacent countries are different colours? 3. Does a potential sudoku problem have a unique solution?

22 Scheduling Exams We first need to associate a graph to the problem

23 Scheduling Exams We first need to associate a graph to the problem The vertices are the classes

24 Scheduling Exams We first need to associate a graph to the problem The vertices are the classes We ll draw an edge between two classes if a student is taking both of them

25 Scheduling Exams We first need to associate a graph to the problem The vertices are the classes We ll draw an edge between two classes if a student is taking both of them We need to assign a time/ colour for each exam

26 Scheduling Exams We first need to associate a graph to the problem The vertices are the classes We ll draw an edge between two classes if a student is taking both of them We need to assign a time/ colour for each exam If there is an edge between two classes, they need a different colour

27 Scheduling Exams We first need to associate a graph to the problem The vertices are the classes We ll draw an edge between two classes if a student is taking both of them We need to assign a time/ colour for each exam If there is an edge between two classes, they need a different colour The question is asking for the minimal number of colours

28 Scheduling Suppose that the associated graph is: COM101 BIO103 MA109 PHI120 MA111 MA123 MA137

29 Scheduling Suppose that the associated graph is: COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 What s the fewest number of required time slots? A: 1 B: 2 C: 3 D: 4

30 Scheduling COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 We can t run all exams at the same time

31 Scheduling COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 We can t run all exams at the same time Can we have just two exam times?

32 Scheduling COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 Com 101, Ma 111, and Phi 120 have to be at different times Can we schedule all exams in three time slots?

33 Scheduling COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 Com 101, Ma 111, and Phi 120 have to be at different times Can we schedule all exams in three time slots?

34 Scheduling COM101 BIO103 MA109 PHI120 MA111 MA123 MA137 Com 101, Ma 111, and Phi 120 have to be at different times Can we schedule all exams in three time slots? Yes

35 Cartography We have a map

36 Cartography We have a map

37 Cartography We have a map We want to colour the countries (and the ocean) so that adjacent regions do not have the same colour

38 Cartography We have a map We want to colour the countries (and the ocean) so that adjacent regions do not have the same colour What is the fewest number of colours that you need? A: 2 B: 3 C: 4 D: 5 E: None of the above

39 Cartography We have a map Here is a colouring with 4 colours

40 Cartography We have a map Here is a colouring with 4 colours We can t colour the map using just 3 colours (consider the three countries in the bottom-right, and the ocean)

41 Cartography We have a map Here is a colouring with 4 colours We can t colour the map using just 3 colours (consider the three countries in the bottom-right, and the ocean) We want to know this number for any map

42 Cartography We have a map Here is a colouring with 4 colours We can t colour the map using just 3 colours (consider the three countries in the bottom-right, and the ocean) We want to know this number for any map We can phrase this question using graphs:

43 Cartography We have a map Here is a colouring with 4 colours We can t colour the map using just 3 colours (consider the three countries in the bottom-right, and the ocean) We want to know this number for any map We can phrase this question using graphs: Countries are vertices If countries are adjacent, we ll draw an edge

44 Cartography We have a map

45 Cartography We have a map Again, we want to colour the vertices so that adjacent vertices have different colour

46 Cartography We have a map A vertex colouring assigns a colour to each vertex so that adjacent vertices have different colours

47 Sudoku Another example of vertex colouring: Sudoku

48 Sudoku Another example of vertex colouring: Sudoku Given a grid such as: we want to enter 1,..., 9 in each cell so that each row, column, and 3 3 square has exactly one of each possible entry

49 Sudoku What should the vertices be?

50 Sudoku What should the vertices be? the small squares

51 Sudoku What should the vertices be? the small squares When should we have an edge between two vertices?

52 Sudoku What should the vertices be? the small squares When should we have an edge between two vertices? when the vertices are in the same column, the same row, or the same 3 3 square

53 Sudoku What should the vertices be? the small squares When should we have an edge between two vertices? when the vertices are in the same column, the same row, or the same 3 3 square A solution is a colouring using 1,..., 9

54 Sudoku What should the vertices be? the small squares When should we have an edge between two vertices? when the vertices are in the same column, the same row, or the same 3 3 square A solution is a colouring using 1,..., 9 We want to know if there is a unique such colouring