Graph Theory

Tyler Zhu

Outline

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Introduction Deﬁnitions A Survey of Graph Theory Food for Thought Eulerian Trails Eulerian Trails Characterization Tyler Zhu Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! Irvington Math Club In Conclusion The Three Utilities Problem In Three January 17, 2018 Dimensions. . . Algebraic Topology

1 / 109 Outline Graph Theory Tyler Zhu Outline Teasers Outline Teasers

Introduction to Graph Theory Introduction Deﬁnitions Deﬁnitions Food for Thought Food for Thought Eulerian Trails Eulerian Trails Paths of Euler and Hamilton Characterization Eulerian Trails Hamiltonian Paths Planar Graphs Characterization Euler Characteristic Let’s Calculate It! Hamiltonian Paths In Conclusion The Three Utilities Problem Planar Graphs In Three Dimensions. . . Euler Characteristic Algebraic Topology Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

2 / 109 Consider the map of Konigsberg colorized, circa 1736, presented below. Find a walk through the city that crosses each of the bridges once and only once.

Figure: Bridges of Kongsberg

Alice, Bob, and Carl live in a two-dimensional village and need to be connected with each of the one sources of heat, water, and electricity. Is there a way to make all nine connections without any of the lines overlapping?

Graph Theory Teasers Graph Theory Tyler Zhu

Outline

Teasers

Introduction Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

3 / 109 Graph Theory Teasers Graph Theory Tyler Zhu Consider the map of Konigsberg colorized, circa 1736, presented below. Find a walk through the city that crosses Outline each of the bridges once and only once. Teasers Introduction Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Figure: Bridges of Kongsberg Algebraic Topology

4 / 109 A graph G consists of vertices or nodes, the points, and edges, the lines. We frequently write V (G) and E(G) for the vertex and edge sets of G respectively, and call |V (G)| and |E(G)| the order and size of the graph respectively.

u x y

Figure: A graph of order 5 and size 6.

Deﬁnitions Graph Theory Tyler Zhu

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Teasers

5 / 109 v

u x y

Figure: A graph of order 5 and size 6.

Deﬁnitions Graph Theory Tyler Zhu A graph G consists of vertices or nodes, the points, and edges, the lines. We frequently write V (G) and E(G) for Outline the vertex and edge sets of G respectively, and call |V (G)| Teasers Introduction and |E(G)| the order and size of the graph respectively. Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

6 / 109 In Figure 6, V (G) = {u, v, w, x, y} and E(G) = {uv, ux, uw, vx, wx, xy}. If e = uv is an edge of G, then u and v are said to be joined by the edge e. In this case, u and v are referred to as neighbors of each other.

Deﬁnitions Graph Theory Tyler Zhu A graph G consists of vertices or nodes, the points, and edges, the lines. We frequently write V (G) and E(G) for Outline the vertex and edge sets of G respectively, and call |V (G)| Teasers Introduction and |E(G)| the order and size of the graph respectively. Deﬁnitions Food for Thought Eulerian Trails v Eulerian Trails Characterization Hamiltonian Paths u x y Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion w The Three Utilities Problem In Three Dimensions. . . Figure: A graph of order 5 and size 6. Algebraic Topology

7 / 109 Deﬁnitions Graph Theory Tyler Zhu A graph G consists of vertices or nodes, the points, and edges, the lines. We frequently write V (G) and E(G) for Outline the vertex and edge sets of G respectively, and call |V (G)| Teasers Introduction and |E(G)| the order and size of the graph respectively. Deﬁnitions Food for Thought Eulerian Trails v Eulerian Trails Characterization Hamiltonian Paths u x y Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion w The Three Utilities Problem In Three Dimensions. . . Figure: A graph of order 5 and size 6. Algebraic Topology

In Figure 6, V (G) = {u, v, w, x, y} and E(G) = {uv, ux, uw, vx, wx, xy}. If e = uv is an edge of G, then u and v are said to be joined by the edge e. In this case, u and v are referred to as neighbors of each other. 8 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and ending at v, where consecutive vertices are neighbors. A u − v trail is a u − v walk in which no edge is traversed more than once. A u − v path is a u − v walk in which no vertex is visited more than once. v

u x y

More Notions Graph Theory Tyler Zhu

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9 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

A u − v walk is a sequence of vertices starting at u and ending at v, where consecutive vertices are neighbors. A u − v trail is a u − v walk in which no edge is traversed more than once. A u − v path is a u − v walk in which no vertex is visited more than once. v

u x y

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. Outline

Teasers

10 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

A u − v trail is a u − v walk in which no edge is traversed more than once. A u − v path is a u − v walk in which no vertex is visited more than once. v

u x y

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

11 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

A u − v path is a u − v walk in which no vertex is visited more than once. v

u x y

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

12 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

u x y

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails A u − v path is a u − v walk in which no vertex is visited Eulerian Trails Characterization more than once. Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

13 / 109 A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails A u − v path is a u − v walk in which no vertex is visited Eulerian Trails Characterization more than once. Hamiltonian Paths Planar Graphs v Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities u x y Problem In Three Dimensions. . . w Algebraic Topology

14 / 109 A u − w trail: u − v − x − u − w A u − w path: u − v − x − w

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails A u − v path is a u − v walk in which no vertex is visited Eulerian Trails Characterization more than once. Hamiltonian Paths Planar Graphs v Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities u x y Problem In Three Dimensions. . . w Algebraic Topology

A u − w walk: u − v − x − u − x − w

15 / 109 A u − w path: u − v − x − w

More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails A u − v path is a u − v walk in which no vertex is visited Eulerian Trails Characterization more than once. Hamiltonian Paths Planar Graphs v Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities u x y Problem In Three Dimensions. . . w Algebraic Topology

A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w

16 / 109 More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. A u − v walk is a sequence of vertices starting at u and Outline ending at v, where consecutive vertices are neighbors. Teasers Introduction A u − v trail is a u − v walk in which no edge is traversed Deﬁnitions more than once. Food for Thought Eulerian Trails A u − v path is a u − v walk in which no vertex is visited Eulerian Trails Characterization more than once. Hamiltonian Paths Planar Graphs v Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities u x y Problem In Three Dimensions. . . w Algebraic Topology

A u − w walk: u − v − x − u − x − w A u − w trail: u − v − x − u − w A u − w path: u − v − x − w 17 / 109 Theorem (The Party Theorem) At any party, there is a pair of people who have the same number of friends present. Theorem (First Theorem of Graph Theory) If G is a graph of size m, then the sum of the degrees of every vertex is equal to 2m.

u x y

deg u + deg v + deg w + deg x + deg y =3+2+2+4+1=2 · 6 = 2m

Food for Thought Graph Theory Tyler Zhu

Outline

Teasers

18 / 109 Theorem (First Theorem of Graph Theory) If G is a graph of size m, then the sum of the degrees of every vertex is equal to 2m.

u x y

deg u + deg v + deg w + deg x + deg y =3+2+2+4+1=2 · 6 = 2m

Food for Thought Graph Theory Tyler Zhu Theorem (The Party Theorem) Outline

At any party, there is a pair of people who have the same Teasers

number of friends present. Introduction Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

19 / 109 v

u x y

deg u + deg v + deg w + deg x + deg y =3+2+2+4+1=2 · 6 = 2m

Food for Thought Graph Theory Tyler Zhu Theorem (The Party Theorem) Outline

At any party, there is a pair of people who have the same Teasers

number of friends present. Introduction Deﬁnitions Theorem (First Theorem of Graph Theory) Food for Thought Eulerian Trails If G is a graph of size m, then the sum of the degrees of Eulerian Trails Characterization every vertex is equal to 2m. Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

20 / 109 deg u + deg v + deg w + deg x + deg y =3+2+2+4+1=2 · 6 = 2m

Food for Thought Graph Theory Tyler Zhu Theorem (The Party Theorem) Outline

At any party, there is a pair of people who have the same Teasers

number of friends present. Introduction Deﬁnitions Theorem (First Theorem of Graph Theory) Food for Thought Eulerian Trails If G is a graph of size m, then the sum of the degrees of Eulerian Trails Characterization every vertex is equal to 2m. Hamiltonian Paths Planar Graphs Euler Characteristic v Let’s Calculate It! In Conclusion The Three Utilities Problem u x y In Three Dimensions. . . Algebraic Topology w

21 / 109 Food for Thought Graph Theory Tyler Zhu Theorem (The Party Theorem) Outline

At any party, there is a pair of people who have the same Teasers

number of friends present. Introduction Deﬁnitions Theorem (First Theorem of Graph Theory) Food for Thought Eulerian Trails If G is a graph of size m, then the sum of the degrees of Eulerian Trails Characterization every vertex is equal to 2m. Hamiltonian Paths Planar Graphs Euler Characteristic v Let’s Calculate It! In Conclusion The Three Utilities Problem u x y In Three Dimensions. . . Algebraic Topology w

deg u + deg v + deg w + deg x + deg y =3+2+2+4+1=2 · 6 = 2m

22 / 109 For a connected graph G, any open trail that contains every edge of G is an Eulerian trail. If G contains a closed Eulerian trail, it is Eulerian.

Theorem A nontrivial connected graph G is Eulerian if and only if every vertex of G has even degree.

With this, we can easily characterize graphs possessing an Eulerian trail.

Eulerian Trails Graph Theory Tyler Zhu

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Teasers

23 / 109 Theorem A nontrivial connected graph G is Eulerian if and only if every vertex of G has even degree.

With this, we can easily characterize graphs possessing an Eulerian trail.

Eulerian Trails Graph Theory Tyler Zhu

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Teasers

24 / 109 With this, we can easily characterize graphs possessing an Eulerian trail.

Eulerian Trails Graph Theory Tyler Zhu

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Teasers

Introduction For a connected graph G, any open trail that contains every Deﬁnitions edge of G is an Eulerian trail. If G contains a closed Food for Thought Eulerian Trails Eulerian trail, it is Eulerian. Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Theorem Euler Characteristic Let’s Calculate It! A nontrivial connected graph G is Eulerian if and only if In Conclusion The Three Utilities every vertex of G has even degree. Problem In Three Dimensions. . . Algebraic Topology

25 / 109 Eulerian Trails Graph Theory Tyler Zhu

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26 / 109 Corollary A connected graph G contains an Eulerian trail if and only if exactly two vertices of G have odd degree. Furthermore, each Eulerian trail of G begins at one of these odd vertices and ends at another. Problem Prove no such walk exists crossing each of the bridges of Konigsberg once.

Figure: Konigsberg, modernized

Characterization Graph Theory Tyler Zhu

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27 / 109 Problem Prove no such walk exists crossing each of the bridges of Konigsberg once.

Figure: Konigsberg, modernized

Characterization Graph Theory Tyler Zhu Corollary Outline A connected graph G contains an Eulerian trail if and only if Teasers

exactly two vertices of G have odd degree. Furthermore, Introduction each Eulerian trail of G begins at one of these odd vertices Deﬁnitions Food for Thought and ends at another. Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

28 / 109 Figure: Konigsberg, modernized

Characterization Graph Theory Tyler Zhu Corollary Outline A connected graph G contains an Eulerian trail if and only if Teasers

exactly two vertices of G have odd degree. Furthermore, Introduction each Eulerian trail of G begins at one of these odd vertices Deﬁnitions Food for Thought and ends at another. Eulerian Trails Eulerian Trails Characterization Problem Hamiltonian Paths Prove no such walk exists crossing each of the bridges of Planar Graphs Euler Characteristic Konigsberg once. Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

29 / 109 Characterization Graph Theory Tyler Zhu Corollary Outline A connected graph G contains an Eulerian trail if and only if Teasers

exactly two vertices of G have odd degree. Furthermore, Introduction each Eulerian trail of G begins at one of these odd vertices Deﬁnitions Food for Thought and ends at another. Eulerian Trails Eulerian Trails Characterization Problem Hamiltonian Paths Prove no such walk exists crossing each of the bridges of Planar Graphs Euler Characteristic Konigsberg once. Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

Figure: Konigsberg, modernized 30 / 109 There is an analog to Eulerian trails. A path in a graph G that contains every vertex of G is called a Hamiltonian path. We studied Eulerian trails because such trails may need to have repeated vertices, unlike paths which necessarily have unique vertices. Unfortunately, these are much less well-behaved. Here is a simple suﬃcient condition for a graph to be Hamiltonian, proven in 1952.

Theorem (Dirac) Let G be a graph of order n ≥ 3. If deg v ≥ n/2 for each vertex v of G, then G is Hamiltonian.

Hamiltonian Paths Graph Theory Tyler Zhu

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Teasers

31 / 109 We studied Eulerian trails because such trails may need to have repeated vertices, unlike paths which necessarily have unique vertices. Unfortunately, these are much less well-behaved. Here is a simple suﬃcient condition for a graph to be Hamiltonian, proven in 1952.

Theorem (Dirac) Let G be a graph of order n ≥ 3. If deg v ≥ n/2 for each vertex v of G, then G is Hamiltonian.

Hamiltonian Paths Graph Theory Tyler Zhu

Outline There is an analog to Eulerian trails. A path in a graph G Teasers that contains every vertex of G is called a Hamiltonian Introduction path. Deﬁnitions Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

32 / 109 Unfortunately, these are much less well-behaved. Here is a simple suﬃcient condition for a graph to be Hamiltonian, proven in 1952.

Theorem (Dirac) Let G be a graph of order n ≥ 3. If deg v ≥ n/2 for each vertex v of G, then G is Hamiltonian.

Hamiltonian Paths Graph Theory Tyler Zhu

33 / 109 Theorem (Dirac) Let G be a graph of order n ≥ 3. If deg v ≥ n/2 for each vertex v of G, then G is Hamiltonian.

Hamiltonian Paths Graph Theory Tyler Zhu

Outline There is an analog to Eulerian trails. A path in a graph G Teasers that contains every vertex of G is called a Hamiltonian Introduction path. Deﬁnitions Food for Thought We studied Eulerian trails because such trails may need to Eulerian Trails have repeated vertices, unlike paths which necessarily have Eulerian Trails Characterization unique vertices. Hamiltonian Paths Planar Graphs Unfortunately, these are much less well-behaved. Here is a Euler Characteristic Let’s Calculate It! simple suﬃcient condition for a graph to be Hamiltonian, In Conclusion The Three Utilities proven in 1952. Problem In Three Dimensions. . . Algebraic Topology

34 / 109 Hamiltonian Paths Graph Theory Tyler Zhu

35 / 109 The mathematician Leonhard Euler noticed something when he calculated the following:

(# of vertices) - (# of edges) + (# of regions)

V − E + R

This number is now known as the Euler Characteristic, and is referred to with the greek letter χ.

Euler Characteristic Graph Theory Tyler Zhu

Outline

Teasers

36 / 109 (# of vertices) - (# of edges) + (# of regions)

V − E + R

This number is now known as the Euler Characteristic, and is referred to with the greek letter χ.

Euler Characteristic Graph Theory Tyler Zhu

Outline

Teasers The mathematician Leonhard Euler noticed something when Introduction Deﬁnitions he calculated the following: Food for Thought Eulerian Trails Eulerian Trails Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

37 / 109 V − E + R

This number is now known as the Euler Characteristic, and is referred to with the greek letter χ.

Euler Characteristic Graph Theory Tyler Zhu

Outline

Teasers The mathematician Leonhard Euler noticed something when Introduction Deﬁnitions he calculated the following: Food for Thought Eulerian Trails Eulerian Trails (# of vertices) - (# of edges) + (# of regions) Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

38 / 109 This number is now known as the Euler Characteristic, and is referred to with the greek letter χ.

Euler Characteristic Graph Theory Tyler Zhu

Outline

Teasers The mathematician Leonhard Euler noticed something when Introduction Deﬁnitions he calculated the following: Food for Thought Eulerian Trails Eulerian Trails (# of vertices) - (# of edges) + (# of regions) Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! V − E + R In Conclusion The Three Utilities Problem In Three Dimensions. . . Algebraic Topology

39 / 109 Euler Characteristic Graph Theory Tyler Zhu

Outline

Teasers The mathematician Leonhard Euler noticed something when Introduction Deﬁnitions he calculated the following: Food for Thought Eulerian Trails Eulerian Trails (# of vertices) - (# of edges) + (# of regions) Characterization Hamiltonian Paths Planar Graphs Euler Characteristic Let’s Calculate It! V − E + R In Conclusion The Three Utilities Problem In Three Dimensions. . . This number is now known as the Euler Characteristic, and Algebraic Topology is referred to with the greek letter χ.

40 / 109 V = 4 V = 3 E = 5 E = 3 R = 3 R = 2 χ = V − E + R χ = V − E + R = 4 − 5 + 3 = 2 = 3 − 3 + 2 = 2