Math 235 Homefun Exercises Dr. G Fall 2017 1: WTF is Discrete Math? It’s time to get you set up with LATEX, a typesetting language that does a much better job of creating professional-looking documents than standard word processors like Microsoft Word. The King’s College Math and Computer Science departments both require their students to become proficient in LATEX. Now is the time to get started! Go to the computer lab, find and run the application TeXstudio, and open the document Welcome to LaTeX.tex that I emailed to you. Compile it by pressing the F5 key. Read the whole thing! 2: Graph Theory Fundamentals 1. Draw a graph with five vertices v1, v2, v3, v4, and v5 such that the degree of v1 is 3, v2 has odd degree, v3 has degree 2, and v4 and v5 are adjacent. 2. For all the graphs you’ve seen so far, go back and count the number of edges in each. Then, add up the degrees of all the vertices. You should see a connection between the number of edges and the sum of all degrees. State that connection clearly and succinctly. 3. If a vertex has an even degree (2, 4, 6, etc.), then we call it an even vertex. Likewise, a vertex having an odd degree is called an odd vertex. Use what you found in the previous exercise to prove that in any graph, the number of odd vertices must be even. 4. Consider a graph with seven vertices. Five vertices have degree 4 and two vertices have degree 2. How many edges does this graph have? 5. Consider a graph whose degree sequence is 5; 5; 4; 4; 3; 3; 3; 3. How many edges does this graph have? 6. What is the degree sequence of the complete graph Kn? 7. How many edges does the complete graph Kn have? Experiment for specific small values of n, and then give the general answer in terms of n. 8. Draw graphs having the following degree sequence, or else explain why no such graph can exist. 1 (a) 1, 1, 1, 1 (b) 1, 1, 1, 1, 1 (c) 2, 2, 2, 2, 2 (d) 4, 4, 3, 2, 2, 0 (e) 3, 3, 3, 3, 2, 2, 2 (f) 3, 3, 3, 3, 3, 3 (g) 6, 6, 4, 2, 2, 2, 1, 1 9. Try to draw a graph where the degrees of the vertices are all different. Can you do it? If so, show me. If not, try to prove why you believe it’s impossible. 10. Imagine the complete graph K6 where each edge has been colored either red or blue. Prove that this graph contains either a red triangle or a blue triangle. 11. Show that K5 can have its edges colored red and blue with no red nor blue triangles. 12. Suppose all vertices in a graph have the same odd degree k. Show that the number of edges is a multiple of k. 13. Show by example that if the vertices in a graph all have the same even degree k, the number of edges may or may not be a multiple of k. 3: Cycles 1.A cut vertex is any vertex in a connected graph that, if deleted, would disconnect the graph. (a) Draw a graph with 5 vertices, one of which is a cut vertex. (b) Draw a graph with 5 vertices such that every vertex is a cut vertex. (c) Explain why no graph having a cut vertex can be Hamiltonian. 2. Suppose that if G is a connected graph and C is a cycle within G. Prove that if one of the edges of C is deleted, G remains connected. 3. A graph G with n vertices is called pancyclic if it contains cycles of every length 3, 4, ..., n. For each of n = 4, 5, and 6, draw a pancyclic graph with n vertices. To make it more interesting, use as few edges as possible. 4. Determine whether the following graphs are Hamiltonian: 5. Prove that any graph with degree sequence 4; 4; 4; 4; 4; 4 is Hamiltonian. 6. The wheel graph Wn consists of a cycle graph Cn together with one extra vertex that is adjacent to all other vertices (so actually, the wheel graph Wn has n + 1 vertices). For example, here is W6: (a) What is the degree sequence of Wn? (b) Prove that Wn is Hamiltonian for all n ≥ 3. 7. Solve the Traveling Salesperson Problem for the weighted graph below two ways: first using the Nearest Neighbor algorithm (start at the top vertex), then using Cheapest Link. 7 1 6 3 4 10 8 2 5 9 8. There are 8 different binary strings of length 3: 000; 100; 010; 001; 110; 101; 011; 111 Draw a graph where each vertex represents one of these strings. Draw an edge between two vertices if and only if the strings differ only in one digit. For example, the strings 100 and 110 differ only in the second digit, so you should draw an edge between these two vertices. Find a Hamilton cycle in this graph. 9. The girth of a graph is the length of its shortest cycle. What is the girth of the complete graph Kn? What is the girth of the cycle graph Cn? 10. Find the girth of the following graph: Note: this graph is famous. It’s called the Petersen Graph. You’ll be seeing it a lot. 4: Circuits 1. Use Fleury’s algorithm to find an Euler circuit in the graph below: 2. Find an Euler circuit in each of the graphs below: G A F B E C D 3. Consider the graph below: (a) Explain why this graph does not possess an Euler circuit. (b) Label the vertices, and then give an Euler path in this graph. 4. For which values of n does the complete graph Kn have an Euler circuit? 5.A bridge is any edge in a connected graph that, if deleted, would disconnect the graph. (a) Draw a graph with 6 vertices having exactly one bridge. (b) Draw a graph with 6 vertices such that every edge is a bridge. (c) Explain why no graph having a bridge can have an Euler circuit. (d) Show by examples that a graph having a bridge may or may not have an Euler path. 6. List all bridges in the following graph: A B G C G D F E Hint: This is actually kinda tricky because of the way the edges are crossing each other. If you try redrawing the graph without any edges crossing, finding the bridges becomes much easier. 7. We saw one practical application of Euler circuits: the Trash Collector Problem. Find at least two other “real world” situations that amount to finding an Euler circuit in some graph. 8. A graph is randomly traceable from a vertex v if, whenever we start from v and traverse the graph in an arbitrary way never retracing any edge, we eventually obtain an Euler circuit. (a) Show that the graph below is randomly traceable from the marked vertex v. v (b) Show that this graph is not randomly traceable from any other vertex. 5: Trees 1. If T is a tree with n vertices, what is the maximum degree any one vertex can have? 2. If T is a tree with n vertices, what is the maximum number of leaves (i.e. vertices of degree 1) T can have? 3. Prove that if T is a tree, then adding any new edge creates a cycle. 4. I’m very bad at chess, so I’ve decided that the next time I play, I will quit after winning a game or after five games have been played. Make a tree showing all possible outcomes. How many outcomes are there? In how many of these outcomes do I win a game? 5. It’s the weekend and I have several ways to spend my Friday and Saturday evenings. I could do some cleaning, watch cat videos on Youtube, or go out with my friends. Make a tree showing all the possible ways I can spend my Friday and Saturday evenings. 6. Draw some trees and count how many edges they have. Then finish this conjecture: If a tree has n vertices, then it has edges. Try to prove your conjecture. 7. Solve the minimal connector problem for this edge-weighted graph. What is the total weight of the spanning tree you found? 3 5 A B C 2 5 5 7 1 9 1 D E F 1 3 6 8 3 G H I 7 4 8. Application to Chemistry: One of the earliest uses of graphs was to enumerate chemical molecules. If we have a molecule consisting only of carbon and hydrogen atoms, then we can represent it as a graph in which each carbon atom appears as a vertex of degree 4 and each hydrogen atom appears as a vertex of degree 1. The graphs of n-butane and 2-methyl propane are shown below. Although they have the same chemical formula C4H10, they are different molecules because the atoms are arranged differently within the molecule. These two molecules form part of a general class of molecules known as the alkanes, or paraffins, with chemical formula CnH2n+2. It is natural to ask how many different molecules there are with this formula. H H C H H H H H H H H C C C C H H C C C H H H H H H H H To find all possible molecules with the formula CnH2n+2, note that the molecule is completely determined by how the carbon atoms are arranged; once the carbon atoms are set, the hydrogen atoms just fill in to bring the degree of each carbon vertex to 4.