Graph Theory Tyler Zhu Outline Teasers 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 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? 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 jV (G)j and jE(G)j the order and size of the graph respectively. v u x y w Figure: A graph of order 5 and size 6. In Figure 6, V (G) = fu; v; w; x; yg and E(G) = fuv; ux; uw; vx; wx; xyg. 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 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 5 / 109 v u x y w Figure: A graph of order 5 and size 6. In Figure 6, V (G) = fu; v; w; x; yg and E(G) = fuv; ux; uw; vx; wx; xyg. 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 jV (G)j Teasers Introduction and jE(G)j 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) = fu; v; w; x; yg and E(G) = fuv; ux; uw; vx; wx; xyg. 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 jV (G)j Teasers Introduction and jE(G)j 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 jV (G)j Teasers Introduction and jE(G)j 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) = fu; v; w; x; yg and E(G) = fuv; ux; uw; vx; wx; xyg. 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 w More Notions 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 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 w More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors. 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 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 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 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 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 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 v u x y w More Notions Graph Theory Tyler Zhu The degree of a vertex v, deg v, is its number of neighbors.