The Meredith graph is a quartic nonhamiltonian graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian.
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-Valent 
-Connected Nonhamiltonian Non-
-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.