Enumeration of Hamiltonian Cycles in Some Grid Graphs Olga Bodroˇ za-Panti´ c, DMI, Faculty of Science, University of Novi Sad, Novi Sad, Serbia Bojana Panti´ c, Ilija Panti´ c, Grammar school “Jovan Jovanovi´ cZmaj”, Novi Sad, Serbia Marija Bodroˇ za-Solarov Institute for Food Technology, University of Novi Sad, Novi Sad, Serbia (Received March 12, 2012) Abstract In polymer science, Hamiltonian paths and Hamiltonian circuits can serve as excellent simple models for dense packed globular proteins. Generation and enumeration of Hamil- tonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate thermodynamics of protein folding. Hamiltonian circuits are a mathemat- ical idealization of polymer melts, too. The number of Hamiltonian cycles on a graph corresponds to the entropy of a polymer system. In this paper, we present new character- izations of the Hamiltonian cycles in a labeled rectangular grid graph P m × P n and in a labeled thin cylinder grid graph C m ×P n . We proved that for any fixed m, the numbers of Hamiltonian cycles in these grid graphs, as sequences with counter n, are determined by linear recurrences. The computational method outlined here for finding these difference equations together with the initial terms of the sequences has been implemented. The generating functions of the sequences are given explicitly for some values of m. The ob- tained data are consistent with data obtained in the works by Kloczkowski and Jernigan, and Schmalz et al. 1. Introduction In polymer science, the functional properties of proteins depend upon their three-dimensio- nal structures, which arise because particular sequences of amino acids in polypeptide chains fold to generate, from linear chains, compact domains with specific structures. For solving so-called the protein folding problem one needs: a) to develop a model of proteins which contains the essentials of the system but which, at the same time, is sufficiently MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 181-204 ISSN 0340 - 6253