The Number of Perfect Matchings in Möbius Ladders and Prisms

R. S. Lekshmi, Douglas B. West

备注：: The authors have been informed that the results in this paper are not new; the sequences for the two problems appear in http://oeis.org/A020878 and http://oeis.org/A068397 , respectively. The results can also be found in J.P. McSorley, Counting structures in the Möbius ladder, Discrete Math. 184 (1998), no. 1-3, 137--164. The result for the prism was known to E. Deutsch in 2004 (unpublished)

英文摘要：

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.

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