By the handshaking lemma, a face-regular polyhedron with an odd number of faces must have faces with an even number of edges, which for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by Broersma, H. J.; Duijvestijn, A. J. W.; Göbel, F. (1993), "Generating all 3-connected 4-regular planar graphs from the octahedron graph", Journal of Graph Theory, 17 (5): 613–620, doi:10.1002/jgt.3190170508, MR1242180. Table 1, p. 619, shows that there is only one with nine faces.
Dillencourt, Michael B. (1996), "Polyhedra of small order and their Hamiltonian properties", Journal of Combinatorial Theory, Series B, 66 (1): 87–122, doi:10.1006/jctb.1996.0008, MR1368518; see Table IX, p. 102.
Biggs, N.L. (1981), "T.P. Kirkman, mathematician", The Bulletin of the London Mathematical Society, 13 (2): 97–120, doi:10.1112/blms/13.2.97, MR0608093.
By the handshaking lemma, a face-regular polyhedron with an odd number of faces must have faces with an even number of edges, which for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by Broersma, H. J.; Duijvestijn, A. J. W.; Göbel, F. (1993), "Generating all 3-connected 4-regular planar graphs from the octahedron graph", Journal of Graph Theory, 17 (5): 613–620, doi:10.1002/jgt.3190170508, MR1242180. Table 1, p. 619, shows that there is only one with nine faces.
Dillencourt, Michael B. (1996), "Polyhedra of small order and their Hamiltonian properties", Journal of Combinatorial Theory, Series B, 66 (1): 87–122, doi:10.1006/jctb.1996.0008, MR1368518; see Table IX, p. 102.
Hosoya, Haruo; Nagashima, Umpei; Hyugaji, Sachiko (1994), "Topological twin graphs. Smallest pair of isospectral polyhedral graphs with eight vertices", Journal of Chemical Information and Modeling, 34 (2): 428–431, doi:10.1021/ci00018a033.
Biggs, N.L. (1981), "T.P. Kirkman, mathematician", The Bulletin of the London Mathematical Society, 13 (2): 97–120, doi:10.1112/blms/13.2.97, MR0608093.
By the handshaking lemma, a face-regular polyhedron with an odd number of faces must have faces with an even number of edges, which for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by Broersma, H. J.; Duijvestijn, A. J. W.; Göbel, F. (1993), "Generating all 3-connected 4-regular planar graphs from the octahedron graph", Journal of Graph Theory, 17 (5): 613–620, doi:10.1002/jgt.3190170508, MR1242180. Table 1, p. 619, shows that there is only one with nine faces.
