We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z d . The expected length of a random self avoiding path is considered. We discuss possible definitions of "critical" behavior in the finite volume setting. We also define a "critical exponent" γ for sequences of graphs of size tending to infinity, and show that γ = 1 in the large girth case.