MATH 513 EXAM I This exam is worth 100 points, with each problem worth 20 points. Please complete Problem 1 and then any four of the remaining problems. Unless indicated, you must justify your answer to receive credit for a solution. When submitting your exam, please indicate which problems you want graded by writing them in the upper right corner on the cover of your exam booklet. You must select exactly four problems; any unselected problems will not be graded, and if you select more than four only the first four (in numerical order) will be graded. (1) Please give the definitions requested. You may assume that we know the definition of graph. (a) Define order of a graph and size of a graph. (b) Define connected graph, and bipartite graph. (c) Define tree and spanning tree. (d) Define the Ramsey number R(p, q). (2) Find the number of spanning trees of the graph H given in Figure 1(a). (Note: you are not required to compute this any particular way.) (3) Find the chromatic polynomial of the graph Kn − e, where e is any edge. (4) For each of the following Prüfer sequences, draw the corresponding labelled trees: (a) 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4 (b) 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2 (c) a, a − 1, . . . , 3, 2, 1 e2k be the graph obtained by taking the cycle C2k and then adding k (5) Let k ≥ 2, and let C e12 ). Show that C e2k is planar if new edges by joining opposite vertices (Figure 1(b) shows C e e e2k−2 as and only if k = 2. (Hint: You know C6 by another name. Now show that C2k has C a subdivision.) (6) Suppose a connected graph G on n vertices has exactly one cycle. How many edges does G have? (7) Let G be a connected graph. Show that if an edge e lies on no cycles of G then e is a bridge. 1 2 3 4 e12 (b) C (a) H Figure 1. Date: Thursday, 18 October 2007.