  There is an extensive literature concerning self-avoiding walk on infinite
graphs, but the subject is relatively undeveloped on finite graphs. The purpose
of this paper is to elucidate the phase transition for self-avoiding walk on
the simplest finite graph: the complete graph. We make the elementary
observation that the susceptibility of the self-avoiding walk on the complete
graph is given exactly in terms of the incomplete gamma function. The known
asymptotic behaviour of the incomplete gamma function then yields a complete
description of the finite-size scaling of the self-avoiding walk on the
complete graph. As a basic example, we compute the limiting distribution of the
length of a self-avoiding walk on the complete graph, in subcritical, critical,
and supercritical regimes. This provides a prototype for more complex unsolved
problems such as the self-avoiding walk on the hypercube or on a
high-dimensional torus.
