Is the graph in Fig. 1 Eulerian? Is it Hamiltonian? Justify your answers.
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If a planar graph has the degree sequence {2;2;3;3;4;4;4}, how many faces will it have? Draw a planar graph with this degree sequence and number the faces to check your answer.
Let t(n) denote the number of ways in which 2n tennis players can be paired to play n matches. Determine a recurrence equation for the sequence t(n).
Give examples of trivially true and vacuously true statements.
Which of the following sentences are statements? Give reasons for your answer.
i) Are the buses running today?
ii) What a pleasant weather!
iii) x2 +1 = 0 for a real value of x.
iv) Every odd number is a prime.
For the graph in Fig. Fig. 2 on the following page find a minimal colouring.
A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?
Show that any tree with exactly two vertices of degree 1 is a path.