all(len(G.longest_path(v)) == len(G) for v in G.vertices())

def homtrac_orbits(g):
    orbits = g.automorphism_group(return_group=False, orbits=True)
    n = len(g)
    return all(len(g.longest_path(o[0])) == n for o in orbits)

def ptfnh(G):
    """Graph property: planar, triangle-free, non-hamiltonian.

    If G has the property, then any subgraph obtained by deleting one edge also has it,
    so this may be used with the Sage graphs generator.
    """
    return G.is_planar() and G.is_triangle_free() and not G.is_hamiltonian()

for n in range(5, 12):
    totct = 0
    for G in graphs(n, ptfnh):
        if not G.is_connected():
            continue
        totct += 1
        if homtrac_orbits(G):
            print G.to_dictionary()
    print n, ':', totct, 'planar connected triangle-free non-hamiltonian graphs'

5 : 5 planar connected triangle-free non-hamiltonian graphs
6 : 15 planar connected triangle-free non-hamiltonian graphs
7 : 51 planar connected triangle-free non-hamiltonian graphs
8 : 210 planar connected triangle-free non-hamiltonian graphs
9 : 1006 planar connected triangle-free non-hamiltonian graphs
10 : 5831 planar connected triangle-free non-hamiltonian graphs
11 : 39210 planar connected triangle-free non-hamiltonian graphs