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The Number of Perfect Matchings in Möbius Ladders and Prisms

The 1970s conjecture of Lovász and Plummer that the number of perfect matchings in any $3$-regular graph is exponential in the number of vertices was proved in 2011 by Esperet, Kardoš, King, Král', and Norine. We give the exact formula for the number of perfect matchings in two families of $3$-regular graphs. In the graph consisting of a $2n$-cycle with diametric chords (also known as the Möbius ladder $M_n$ and a Harary graph) and in the cartesian product of the cycle $C_n$ with an edge (called the cycle prism), the number of matchings is the sum of the Fibonacci numbers $F_{n-1}$ and $F_{n+1}$, plus two more for the Möbius ladder when $n$ is odd and for the cycle prism when $n$ is even.