Geodesic
On the Number of 2-factors in Rectangular Lattice Graphs
Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 23 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $f_m(n)$ and $h_m(n)$ denote the number of 2-factors and the number of connected 2-factors (Hamiltonian cycles) respectively in a $(m-1)\times(n-1)$ grid i.e., in the labelled graph $P_m\times P_n$. We show that for each fixed $m$ ($m>1$) the sequences $f_m=(f_m(2),f_m(3), \dots)$ and $h_m=(h_m(2),h_m(3),\dots)$ satisfy difference equations (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding these difference equations together with the initial terms of the sequence. The generating functions of $f_m$ and $h_m$ are rational functions $\Cal F_m$ and $\Cal H_m$ respectively, and they are given explicitly for some values of $m$.
Classification : 05C70 05C45 
@article{PIM_1994_N_S_56_70_a3,
     author = {Olga Bodro\v{z}a-Panti\'c and Ratko To\v{s}i\'c},
     title = {On the {Number} of 2-factors in {Rectangular} {Lattice} {Graphs}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {23 },
     publisher = {mathdoc},
     volume = {_N_S_56},
     number = {70},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1994_N_S_56_70_a3/}
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Olga Bodroža-Pantić; Ratko Tošić. On the Number of 2-factors in Rectangular Lattice Graphs. Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 23 . http://geodesic.mathdoc.fr/item/PIM_1994_N_S_56_70_a3/