Eulerian and Hamiltonian Graphs · Superchallenge problem

The Petersen graph has $10$ vertices arranged in two rings: an inner pentagon $0,1,2,3,4$ where each is connected to its two neighbors, and an outer pentagon $5,6,7,8,9$ (similarly connected), plus five spokes connecting $i$ to $i+5$ for $i=0,\ldots,4$. Consider two adjacent vertices in this graph, say $0$ and $1$ (connected by an edge). How many distinct Hamiltonian paths start at vertex $0$ and end at vertex $1$ (the other endpoint of their shared edge)?