Counting Claw-Free Cubic Graphs

Let Hn be the number of claw-free cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for {Hn}. This leads to a recurrence relation for Hn which shows {Hn} to be P -recursive and which enables the sequence to be computed efficiently. Thus the enumeration of labeled claw-free cubic graphs can be added to the handful of known counting problems for regular graphs with restrictions which have been proved P -recursive.