Counting Hamiltonian cycles on planar random lattices

Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles is obtained and its singularity is studied. Hamiltonian cycles have often been used to model collapsed polymer globules[1]. A Hamiltonian cycle of a graph is a closed path which visits each of the vertices once and only once. The number of Hamiltonian cycles on a lattice corresponds to the entropy of polymer system on it in collapsed but disordered phase. There can be no polynomial time algorithms to determine whether a graph admits a Hamiltonian cycle or not which work for arbitrary graphs[2]. Even for regular lattices, the number of Hamiltonian cycles are not obtained exactly except for a few well-behaved lattices[3, 4, 5, 6] In the present work, I consider the problem of counting the number of Hamiltonian cycles on planar random lattices, or planar random fat graphs. I obtain the exact generating function for the number. Let S be the set of all planar trivalent fat graphs with n vertices possibly with multiple edges and self-loops. Graphs that are isomorphic are identified. The set S̃ is the labeled version of S, namely, vertices of G̃ ∈ S̃ are labeled as 1, . . . , n and G̃1, G̃2 ∈ S̃ n are considered identical only if a graph isomorphism preserves labels. The symmetric group of degree n naturally acts on S̃ by label permutation. The automorphism group Aut G is the stabilizer subgroup of G̃. A Hamiltonian cycle of a labeled graph G̃ ∈ S̃ is a directed closed path ( sequence of consequential distinct edges) which visits each of n vertices once and only once. Hamiltonian cycles are understood as furnished with a direction and a base point. The number of Hamiltonian cycles of G̃ is denoted by H(G) because it is independent of the way of labeling. See figure 1 for an example. e-mail: [email protected]