Tietze's Graph

Tietze's graph is the cubic graph on 12 nodes and 18 edges illustrated above in a number of embeddings.

It is implemented in the Wolfram Language as GraphData["TietzeGraph"].

Tietze's graph is the unique almost Hamiltonian cubic graph on 12 vertices (Punnim et al. 2007). In fact, it is also maximally nonhamiltonian (Clark and Entringer 1983).

Tietze's graph provides a 6-color coloring of the Möbius strip as illustrated above.

The plots above show the adjacency, incidence, and graph distance matrices for Tietze's graph.

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976.

Clark, L. and Entringer, R. "Smallest Maximally Nonhamiltonian Graphs." Periodica Math. Hungarica 14, 57-68, 1983.

Punnim, N.; Saenpholphat, V.; and Thaithae, S. "Almost Hamiltonian Cubic Graphs." Int. J. Comput. Sci. Netw. Security 7, 83-86, 2007.