2 hours THE UNIVERSITY OF MANCHESTER. 6 June :45 11:45

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1 2 hours THE UNIVERSITY OF MANCHESTER DISCRETE MATHEMATICS 6 June :45 11:45 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. University approved calculators may be used 1 of 6 P.T.O.

2 SECTION A Answer ALL 3 questions A1. (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 3 are contained in the digraph G(V, E) below? v 1 v 2 v 5 v 4 v 3 (c) How many spregs with distinguished vertex v 3 that contain the cycle induced by {v 1, v 2, v 4 } appear the graph above? [10 marks] A2. This question concerns ways to characterize the growth of a function f : N R + in terms of a second function g : N R +. (a) Explain what is meant by the notations f(n) = O (g(n)), f(n) = Ω (g(n)) and f(n) = Θ (g(n)). (b) Prove the following: if f 1 (n) = Ω (g 1 (n)) and f 2 (n) = Ω (g 2 (n)), then (f 1 (n) f 2 (n)) = Ω (g 1 (n) g 2 (n)); if f(n) = n + ln(n) then f(n) = Θ(n). [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 10 None B 12 None C 15 None D 6 A & C E 3 A & B F 5 B & C G 7 D & F H 6 D & E I 9 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. (a) Explain what is meant by the following: the degree sequence of a graph G(V, E); the statement that two graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) are isomorphic; the statement that G(V, E) is a tree. (b) Prove the following statement or find a counterexample: If two graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) both have degree sequence {1, 1, 1, 2, 2, 3} then they are isomorphic. If you offer a counterexample, be sure to prove that your two graphs aren t isomorphic. The remainder of the question concerns sequences of positive integers d j N of the form D = {d 1, d 2,..., d n }, ordered so that 0 < d 1 d 2 d n. Prove the following statements or give a counterexample. (c) If D is the degree sequence of a tree on n vertices then it satisfies n j=1 d j = 2(n 1). (d) If D is the degree sequence of a tree on n 3 vertices then d 1 = d 2 = 1 and d n = max j (d j ) > 1. (e) If n j=1 d j = 2(n 1) then there exists a tree on n vertices whose degree sequence is D. [25 marks] 4 of 6 P.T.O.

5 B5. (a) Explain what is meant by: a cycle in a graph G(V, E); the statement that a graph G(V, E) is bipartite; the adjacency matrix of a graph G(V, E); the statement that a graph G(V, E) is Eulerian; the statement that a graph G(V, E) is Hamiltonian. (b) Prove that if G is a bipartite graph, then every cycle has even length. (c) Consider a graph G whose adjacency matrix is Is G Eulerian? Is it Hamiltonian? Justify your answers rigorously. The remainder of the question concerns the graph H illustrated below. v 1 v 2 v 3 v 6 v 5 v 4 (d) How many walks of length 3 start at v 1 and finish at v 2? (e) Define A to be the adjacency matrix of H, and consider the sequence A k of matrix powers of A defined recursively by A k+1 = AA k and A 1 = A. Prove that A k contains at least one zero entry for all k N. [25 marks] 5 of 6 P.T.O.

6 B6. Parts (b), (d) and (e) refer to the graph below. v 3 v 7 v 1 v 4 v 6 v 2 v 5 v 8 (a) For a graph G(V, E), explain what is meant by: a k-colouring of a graph G(V, E); the chromatic number χ(g) of a graph; an induced subgraph of G; the statement that G is homeomorphic to a graph H(V, E ). (b) Find the chromatic number of the graph pictured above, justifying your answer rigorously. (c) Suppose that every student at a school is studying one of the following combinations of subjects and that at least one student is pursuing each combination: Maths, Physics, French Maths, English, German English, French, Physics Chemistry, Physics, French What is the minimum number of examination periods required for exams in the specified courses so that the students involved have no exam clashes? Justify your answer. (d) Find an induced subgraph of the graph pictured above that is homeomorphic, but not isomorphic, to K 4. (e) Is the graph pictured above planar? Justify your answer. [25 marks] END OF EXAMINATION PAPER 6 of 6