Q4 (10 points)
A complete bipartite graph $K_{m,n} = (V, E)$ is a graph defined
by
$V = [m + n]$
and
$E = \{\{(a, b)\} : 1 \le a \le m \text{ and } m + 1 \le b \le m + n\}$.
Show that the number of ways of colouring this graph in (at
most) $k$-colours is
$\sum_{i+j \le k} S(m, i)i!S(n, j)j! \binom{k}{i, j, k - i - j}$
where $S(N, R)$ denotes the number of ways of distributing $N$
distinct objects into $R$ nonempty indistinguishable sets.
Hint: How is the number of surjective functions from
$[n] \to [r]$ related to $S(n, r)$, the Stirling number?