Review CS311H: Discrete Mathematics Introduction to Graph Theory I What are properties of simple graphs? I What is the Handshaking Theorem? Instructor: I¸sıl Dillig I What is the induced subgraph of G on vertices V ? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 1/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 2/29 Complete Graphs Bipartite graphs I A complete graph is a simple undirected graph in which every I A simple undirected graph G = (V , E) is called bipartite if V pair of vertices is connected by one edge. can be partitioned into two disjoint sets V1 and V2 such that every edge in E connects a V1 vertex to a V2 vertex I Complete graph with n vertices denoted Kn . C A D EB I How many edges does a complete graph with n vertices have? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 3/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 4/29 Examples Bipartite and Non-Bi-partite Graphs Questions about Bipartite Graphs I Is this graph bipartite? A I Does there exist a complete graph that is also bipartite? B C I What about this graph? A E I Consider a graph G with 5 nodes and 7 edges. Can G be C bipartite? D B F Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 5/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 6/29 1 Graph Coloring Question I A coloring of a graph is the assignment of a color to each vertex so that no two adjacent Consider a graph G with vertices v1, v2, v3, v4 and edges vertices are assigned the same color. { } (v1, v2), (v1, v3), (v2, v3), (v2, v4). I A graph is k-colorable if it is possible to color it using k colors. Which of the following are valid colorings for G? I e.g., graph on left is 3-colorable 1. v1 = red, v2 = green, v3 = blue I Is it also 2-colorable? 2. v1 = red, v2= green, v3 = blue, v4 = red I The chromatic number of a graph is the least number of 3. v1 = red, v2= green, v3 = red, v4 = blue colors needed to color it. I What is the chromatic number of this graph? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 7/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 8/29 Examples Applications of Graph Coloring I Graph coloring has lots of applications, particularly in scheduling. What are the chromatic numbers for these graphs? A B A B A B I Example: What’s the minimum number of time slots needed so that no student is enrolled in conflicting classes? C D C D C D 311 314 312 331 429 439 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 9/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 10/29 A Scheduling Problem Bipartite Graphs and Colorability Prove that a graph G = (V , E) is bipartite if and only if it is I The math department has 6 committees C1,..., Cn that 2-colorable. meet once a month. I The committee members are: C = Allen, Brooks, Marg C = Brooks, Jones, Morton 1 { } 2 { } C = Allen, Marg, Morton C = Jones, Marg, Morton 3 { } 4 { } C = Allen, Brooks C = Brooks, Marg, Morton 5 { } 6 { } I How many different meeting times must be used to guarantee that no one has conflicting meetings? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 11/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 12/29 2 Complete graphs and Colorability Degree and Colorability Theorem: Every simple graph G is always max degree(G) + 1 Prove that any complete graph Kn has chromatic number n. colorable. I Proof is by induction on the number of vertices n. I Let P(n) be the predicate “A simple graph G with n vertices is max-degree(G)-colorable” I Base case: n = 1. If graph has only one node, then it cannot have any edges. Hence, it is 1-colorable. I Induction: Consider a graph G = (V , E) with k + 1 vertices I Consider arbitrary v V with neighbors v ,..., v ∈ 1 n Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 13/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 14/29 Degree and Colorability, cont. Degree and Colorability, cont. I Either cp+1 is (i) new or (ii) already used by C I Remove v and all its incident edges from G; call this G0 I Case 1: If already used, G is max degree(G0) + 1-colorable, I By the IH, G0 is max-degree(G0) + 1 colorable therefore also max degree(G) + 1-colorable I Let C 0 be the coloring of G0: Suppose C 0 assigns colors c ,..., c to v’s neighbors. Clearly, p n. I Case 2: Coloring C uses p + 1 colors 1 p ≤ I We know p n where n is the number of v’s neighbors I Now, create coloring C for G: ≤ I C (v 0) = C 0(v 0) for any v = v 0 6 I What can we say about max degree(G)? I C (v) = cp+1 I Thus, G is max degree(G) + 1-colorable Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 15/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 16/29 Star Graphs and Colorability Question About Star Graphs Suppose we have two star graphs Sk and Sm . Now, pick a random vertex from each graph and connect them with an edge. I A star graph Sn is a graph with one vertex u at the center and the only edges Which of the following statements must be true about the are from u to each of v1,..., vn 1. resulting graph G? − I Draw S2, S3, S4, S5. 1. The chromatic number of G is 3 I What is the chromatic number of Sn ? 2. G is 2-colorable. 3. max degree(G) = max(k, m). Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 17/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 18/29 3 Connectivity in Graphs Paths a I A path between u and v is a sequence of edges that starts at vertex u, moves along b I Typical question: Is it possible to get from adjacent edges, and ends in v. some node u to another node v? x I Example: u, x, y, w is a path, but u, y, v and w I Example: Train network – if there is path u, a, x are not from u to v, possible to take train from u to u y v and vice versa. I Length of a path is the number of edges v traversed, e.g., length of u, x, y, w is 3 d I If it’s possible to get from u to v, we say u and v are connected and there is a path I A simple path is a path that does not repeat between u and v any edges c I u, x, y, w is a simple path but u, x, u is not Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 19/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 20/29 Example Connectedness I Consider a graph with vertices x, y, z, w and edges { } I A graph is connected if there is a path (x, y), (x, w), (x, z), (y, z) between every pair of vertices in the graph I What are all the simple paths from z to w? I Example: This graph not connected; e.g., no path from x to d I What are all the simple paths from x to y? I A connected component of a graph G is a I How many paths (can be non-simple) are there from x to y? maximal connected subgraph of G Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 21/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 22/29 Example Circuits I A circuit is a path that begins and ends in the same vertex. I Prove: Suppose graph G has exactly two vertices of odd degree, say u and v. Then G contains a path from u to v. I u, x, y, x, u and u, x, y, u are both circuits I A simple circuit does not contain the same I edge more than once I I u, x, y, u is a simple circuit, but u, x, y, x, u is not I I Length of a circuit is the number of edges it I contains, e.g., length of u, x, y, u is 3 I In this class, we only consider circuits of length 3 or more Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 23/29 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Introduction to Graph Theory 24/29 4 Cycles Example I Prove: If a graph has an odd length circuit, then it also has an a odd length cycle. b I Huh? Recall that not every circuit is a a cycle. I A cycle is a simple circuit with no repeated x vertices other than the first and last ones.