2 hours THE UNIVERSITY OF MANCHESTER. 23 May :45 11:45

Transcription

1 2 hours MAT20902 TE UNIVERSITY OF MANCESTER DISCRETE MATEMATICS 23 May :45 11:45 Answer ALL TREE questions in Section A (30 marks in total). Answer TWO of the TREE questions in Section B (50 marks in total). If more than TWO questions from Section B are attempted, then credit will be given for the best TWO answers. University approved calculators may be used 1 of 6 P.T.O.

2 SECTION A MAT20902 Answer ALL 3 questions A1. Given a graph G(V, E), say what is meant by: the degree of a vertex v V ; the statement that G is a tree. A saturated hydrocarbon is a molecule with chemical formula C m n (that is, it contains m atoms of carbon and n of hydrogen) in which, as in the examples below, every carbon atom C has four bonds; every hydrogen atom has one bond; there are no double, triple or quadruple bonds and no sequence of bonds forms a cycle. Prove the following results: C C C C If there is a saturated hydrocarbon with formula C m n, then n = 2m + 2. If m is a positive integer and n = 2m + 2, then there exists an example of a saturated hydrocarbon with formula C m n. C C [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) ow many spanning arborescences rooted at v 2 are contained in the digraph below? v 4 v 3 v 1 v 2 (c) Compute the number of spregs with distinguished vertex v 2 that are subgraphs of the digraph above and sketch them all. [10 marks] 2 of 6 P.T.O.

3 MAT20902 A3. The first column in the table below lists various tasks required for the completion of a certain project. The second column gives the time needed to complete each task, while the third column gives each task s immediate prerequisites. Task Time Prerequisites A 4 None B 5 A C 8 B D 6 A E 1 B & D F 7 B & D G 4 C, 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 time (measured from the start of the project) at which it could start and the latest time 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 MAT20902 Answer 2 of the 3 questions B4. (a) Say what is meant by the following terms: a bipartite graph; a amiltonian cycle; a k-colouring of a graph G(V, E); the chromatic number χ(g) of a graph G(V, E). (b) Prove that a connected graph G is a bipartite graph if and only if it has chromatic number χ(g) = 2. (c) Prove that if G is a bipartite graph, then every cycle has even length. The remainder of the question concerns the graph below it is called the erschel graph. (d) What is χ()? Justify your answer rigorously. (e) Does have a amiltonian cycle? Justify your answer rigorously. [25 marks] 4 of 6 P.T.O.

5 MAT20902 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 terms walk, trail and path in a directed graph G(V, E); the statement that a vertex b is reachable from a vertex a in a directed graph G(V, E); 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 7 ] 3,1 in the 7th matrix power of A; the entry [A 10 ] 6,4 in the 10th matrix power of A. The matrix W below tabulates the edge-weights for a certain weighted, directed graph where W ij is the weight of the edge (v i, v j ) and an infinite weight means that the corresponding edge is absent. 2 3 W = 1 (B5.1) 2 1 (d) Explain what is meant by the operations a b and a b in min-plus algebra. (e) 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 MAT20902 B6. (a) Given a connected graph G(V, E) explain what is meant by the following statements: G is amiltonian G has an Eulerian tour G is planar G has girth g Recall that for N 2, the triangular graph T N has vertices corresponding to the two-element subsets of an underlying set with N elements. The edge set of T N includes an edge between two vertices if and only if the corresponding two-element subsets have a non-empty intersection. Thus if the underlying four-element set is {1, 2, 3, 4}, then T 4 is illustrated below. {2, 4} {2, 3} {1, 2} {1, 4} {1, 3} {3, 4} (b) Prove that each vertex of T N has degree 2N 4. (c) Answer the following questions which concern T 6, and note that T 6 is not the graph illustrated above. Support your answers with rigorous arguments: you may use any theorem from the lectures or problem sets without providing a proof. Is T 6 Eulerian? Is T 6 amiltonian? What is the girth of T 6? Is T 6 planar? Does T 6 contain a subgraph isomorphic to K 5? [25 marks] END OF EXAMINATION PAPER 6 of 6