In a network with three graph edges at each graph vertex, the number of Hamiltonian cycles through a specified graph edge is 0 or even.
Smith's Network Theorem
See also
Graph Edge, Hamiltonian Cycle, NetworkExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Smith's Network Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmithsNetworkTheorem.html