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1 2 hours THE UNIVERSITY OF MANCHESTER DISCRETE MATHEMATICS 22 May :00 16:00 Answer ALL THREE questions in Section A (30 marks in total) and TWO of the THREE questions in Section B (50 marks in total). If all THREE questions from Section B are attempted, then credit will be given for the TWO best answers. The use of electronic calculators is permitted provided they cannot store text. 1 of 6 P.T.O.

2 SECTION A Answer ALL 3 questions A1. (a) Explain what is meant by the statement that two graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) are isomorphic. (b) Are the two graphs illustrated below isomorphic? Support your answer with a rigorous argument. [10 marks] A2. (a) Say what is meant by the following terms: a spanning arborescence rooted at v in a digraph G(V, E); a single predecessor graph (spreg) with distinguished vertex v in a digraph G(V, E). (b) How many spanning arborescences rooted at v 4 are contained in the digraph below? v 4 v 3 v 1 v 2 (c) Compute the number of spregs with distinguished vertex v 4 that are subgraphs of the digraph above and sketch them all. [10 marks] 2 of 6 P.T.O.

3 A3. The first column in the table below lists various tasks required to complete a certain project. The second column gives the time (in days) required to complete each task, while the third column gives each task s immediate prerequisites. Task Time Prerequisites A 5 None B 6 A C 7 A D 10 B E 8 B & C F 7 C G 6 D, E & F (a) Draw a suitable directed graph representing the project. (b) By finding a critical path through the graph from part (a), find the shortest amount of time in which the project can be completed. (c) For each task, find both the earliest date (measured from the start of the project) on which it could start and the latest date by which it must start if the project is to be completed in minimal time. [10 marks] 3 of 6 P.T.O.

4 SECTION B Answer 2 of the 3 questions B4. While working on this question you may use any theorem from the lectures or problem sets without providing a proof. (a) Given a connected graph G(V, E) explain what is meant by the following statements: G is Hamiltonian; G has an Eulerian tour; G is planar; G is cyclic; G has girth g. The remainder of the question concerns a graph G(V, E) that has V = 10 vertices and E = 41 edges. (b) Answer the following questions, supporting each answer with a rigorous argument. Is G Eulerian? Is G Hamiltonian? What is the girth of G? Is G planar? Does G contain a subgraph isomorphic to K 5? [25 marks] 4 of 6 P.T.O.

5 B5. Parts (b) and (c) of the question refer to the directed graph G below. v 1 v 2 v 6 v 5 v 4 v 3 v 7 (a) Explain what is meant by the following: the statement that two vertices a and b in a directed graph G(V, E) are strongly connected; the adjacency matrix of a directed graph G(V, E). (b) Prove that strong connectedness of vertices is an equivalence relation on the vertex set of a directed graph and find the strongly connected components of the directed graph G above. (c) Using the vertex numbering illustrated above, construct the adjacency matrix A of the directed graph G above and compute: the entry [A 12 ] 7,7 in the 12th matrix power of A; the entry [A 15 ] 7,1 in the 15th matrix power of A. The matrix W below tabulates the edge-weights for a certain weighted, directed graph H where W ij is the weight of the edge (v i, v j ) and an infinite weight means that the corresponding edge is absent. 1 1 W = 3 3 (B5.1) 2 (d) Draw a diagram of H that includes the edge-weights, then answer the following questions, providing rigorous justification for your answers. Do the edge-weights listed in Eqn. (B5.1) specify a metric on the vertex set of H? Is the function well-defined? d(i, j) = weight of a lowest-weight path from v i to v j (e) Explain what is meant by the operations a b and a b in min-plus algebra. (f) Recall that the min-plus powers of a matrix W are defined by W (k+1) = W k W, then compute the least entry in W 5, where W is the weight matrix from Eqn. (B5.1). [25 marks] 5 of 6 P.T.O.

6 B6. (a) Explain what is meant by the following terms: a k-colouring of a graph G(V, E); the chromatic number χ(g) of a graph G(V, E). (b) Say that a graph G(V, E) has maximal degree (G). That is, define (G) = max v V deg(v). Prove χ(g) (G) + 1 and give an example of a graph in which χ(g) = (G) + 1. (c) The keeper of The San Diego Zoo has been thinking of rearranging the exhibits so that animals are shown living together in their natural habitats. Unfortunately, she cannot combine animals arbitrarily as some of them prey on others. The table below summarizes these restrictions, indicating with a dot pairs of animals that should not be housed together. Determine the smallest number of enclosures needed to house all the animals safely and provide a justification for your answer. a b c d e f g h i j a b c d e f g h i j The remainder of the question concerns interval graphs. The vertex set of such a graph consists of a collection of intervals I j R while the edge set includes a pair (I j, I k ) if and only if the corresponding intervals have a nonempty intersection. The Figure below shows the three intervals I 1 = [ 1, 1], I 2 = [0, 2] and I 3 = [1/2, 5/2] as well as the associated interval graph, which is isomorphic to the complete graph K 3. I 3 I 2 I 1 I I 1 I 3 (d) Find the chromatic number of the interval graph whose vertex set consists of the intervals I 1 = [ 1, 2], I 2 = [1, 4] and I 3 = [3, 5]. (e) Find four intervals I 1, I 2, I 3 and I 4 such that the corresponding interval graph has χ(g) = 4. END OF EXAMINATION PAPER 6 of 6 [25 marks]

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