Enumeration of Hamiltonian cycles on a thick grid cylinder - part I: Non-contractible Hamiltonian cycles

Abstract

In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph P m+1 × C n , where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs which enables us to prove that their numbers h nc m (n) satisfy a recurrence relation for every fixed m. From the computational data, we conjecture that the coefficient for the dominant positive characteristic root in the explicit formula for h nc m (n) is 1.

Enumeration of Hamiltonian cycles on a thick grid cylinder - Part II: Contractible Hamiltonian cycles

“…Determining and enumerating Hamiltonian cycles in some specic grid graphs (such as thick grid cylinder graphs, which are studied here) is of quite some relevance to statistical physics [6] and polymer science [2]. An ample amount of references related to this topic may be found in Part I [1]. A few novel applications of this type of research can be found within the eld of network systems, which revolves around computer network functionality.…”

“…The rst kind, denoted by HC nc 's, are not contractible when perceived as closed Jordan curves (see Figure 1b) on the innite cylindrical surface on which the graph P m+1 × C n is settled. They were examined in Part I [1] of this series. The second kind of HC's, denoted by HC c 's, are the contractible ones.…”

“…Further, let us be reminded of a few additional denitions from Part I [1] needed hereinafter. All the rest may as well be found in the said paper, unless explicitly stated dierently.…”

“…Recall from [1] that G m represents an innite grid graph with vertices from the set {(i, j) ∈ Z 2 | 0 ≤ j ≤ m}, in which the square cell determined by the points: (j − 1 + kn, m − i), (j + kn, m − i), (j + kn, m − i + 1) and (j − 1 + kn, m − i + 1), with 1 ≤ i ≤ m and 1 ≤ j ≤ n, is labeled w k ij and is called a window, too. We also say that w k ij belongs to the…”

Counting Hamiltonian Cycles in 2-Tiled Graphs

“…Their method was extended to arbitrarily large grids by Bodroža-Pantić et al [14] and by Stoyan and Strehl [15]. Later, Bodroža-Pantić et al gave some explicit generating functions for the number of Hamiltonian cycles in graphs P m P n and C m P n [16,17]. Earlier, Saburo developed a field theoretic approximation of the number of Hamiltonian cycles in graphs C m C n in [18] as well as in planar random lattices [19].…”