  We prove that, for the edge ideal of a cactus graph, the arithmetical rank is
bounded above by the sum of the number of cycles and the maximum height of its
associated primes. The bound is sharp, but in many cases it can be improved.
Moreover, we show that the edge ideal of a Cohen-Macaulay graph that contains
exactly one cycle or is chordal or has no cycles of length 4 and 5 is a
set-theoretic complete intersection.
