Sample questions

1.

How many integral solutions of

x₁+x₂+x₃+x₄ = 30

Satisfy $x_1 \geq 2$, $x_2\geq 0$, $x_3 \geq -5$, and $x_4 \geq 8$.

2.

How many ways are there to colour the vertices of the 5-cycle so that adjacent vertices receive different colours?

3.

Let $A(X) = \sum_{n=0}^{\infty} a_nX^n$, where a_n=3a_n-1-2a_n-2+2, and a₀=a₁=1. Write A(X) as the quotient of two polynomials.

4.

Show that every automorphism of a tree must fix a vertex or an edge.

5.

Show that there are at most 5 connected simple planar graphs in which every face has the same degree $\phi$ and every vertex has the same degree $\delta > 2$.

6.

For a given graph G show that the chromatic number $\chi(G)$is less than or equal to the maximum degree $\Delta(G)$.

7.

Construct a cyclic Steiner Triple system of order 13.

8.

Construct 2 idempotent mutually orthogonal Latin squares of order 4.

9.

Show that a transversal design on nk points with k groups is equivalent to k-2 mutually orthogonal Latin squares of order n.