A conjecture on the number of Hamiltonian cycles on thin grid cylinder graphs

Abstract

Graph Theory
International audience
We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm×Pn+1C_m \times P_{n+1}. We distinguish two types of Hamiltonian cycles, and denote their numbers hmA(n)h_m^A(n) and hmB(n)h_m^B(n). For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for m≤10m\leq10. The computational data we gathered suggests that hmA(n)∼hmB(n)h^A_m(n)\sim h^B_m(n) when m is even.

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A conjecture on the number of Hamiltonian cycles on
thin grid cylinder graphs
Olga Bodroža-Pantić, Harris Kwong, Milan Pantić
To cite this version:
Olga Bodroža-Pantić, Harris Kwong, Milan Pantić. A conjecture on the number of Hamiltonian cycles
on thin grid cylinder graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2015,
Vol. 17 no. 1 (in progress) (1), pp.219–240. �hal-01196857�

Discrete Mathematics and Theoretical Computer Science DMTCS vol. 17:1, 2015, 219–240
A Conjecture on the Number of Hamiltonian
Cycles on Thin Grid Cylinder Graphs
Olga Bodroˇ
za-Panti´
c1∗Harris Kwong2†Milan Panti´
c3‡
1Dept. of Math. & Info., Faculty of Science, University of Novi Sad, Serbia
2Dept. of Math. Sci., SUNY at Fredonia, NY, U.S.A.
3Dept. of Physics, Faculty of Science, University of Novi Sad, Serbia
received 16th Mar. 2014,revised 27th Jan. 2015, 25th Mar. 2015,accepted 25th Mar. 2015.
We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm×Pn+1 . We distinguish two
types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers hnc
m(n)
and hc
m(n). For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We
derive their generating functions and other related results for m≤10. The computational data we gathered suggests
that hnc
m(n)∼hc
m(n)when mis even.
Keywords: Hamiltonian cycles, generating functions, thin grid cylinder, contractible curves.
1 Introduction
A Hamiltonian path of a simple graph is a path that visits each vertex exactly once. A closed Hamiltonian
path is called a Hamiltonian cycle or Hamiltonian circuit, which we shall abbreviate as HC. The enumer-
ation of Hamiltonian cycles on rectangular grid graphs Pm×Pnhad been studied extensively in, among
others, [2, 4, 9, 15, 10, 13, 14, 17, 19, 20]. In contrast, little work [2, 9, 11, 17] was devoted to enumerate
Hamiltonian cycles on rectangular grid cylinders Cm×Pn.
In this paper we investigate, for each fixed m≥2, the generation and enumeration of Hamiltonian
cycles on Cm×Pn+1, where n≥1. Since ngrows while mis fixed, such graphs are called thin grid
cylinders in the literature. In [2], vertices were encoded. We adopt a different approach by coding the
cells or squares on the cylindrical surface, along with the so-called k-SIST equivalence relation. This
equivalence relation was formerly called k-SISET, and was first used in [4] to enumerate Hamiltonian
cycles on Pm×Pn. A very similar approach for the same enumeration was implemented in [19] using
the language of finite automa.
∗Email: bodroza@dmi.uns.ac.rs. Research supported by the Ministry of Education and Science of the Republic of Serbia
(Grants OI 174018, and III 46005).
†Email: kwong@fredonia.edu.
‡Email: mpantic@df.uns.ac.rs. Research supported by the Ministry of Education and Science of the Republic of Serbia
(Grants OI 171009).
1365–8050 c
2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

220 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
We distinguish two different types of HCs. In the sense of homotopy: one type of HCs are contractible
(as Jordan curves) to a point, and the other type of HCs are not. We denote them HCcand HCnc, respec-
tively. Simply put, a HCnc is one that “perches” on or wraps around the cylinder likes a bracelet on an
arm, and a HCccan be “pasted” on the cylindrical surface. Let hnc
m(n)and hc
m(n)be the number of HCncs
and HCcs, respectively, on Cm×Pn+1 , Our objective is to determine, for each fixed m, the sequences
hnc
n={hnc
m(n)}n≥1and hc
m={hc
m(n)}n≥1. It is obvious that the number of HCs on Cm×Pn+1 is
given by hm(n) = hnc
m(n) + hc
m(n).
We characterize both types of HCs, and use it to define, for each fixed m≥2, a digraph Dm. The
original enumeration problem is equivalent to counting oriented walks of length n−1in this digraph
with first and last vertices from two special sets. Using the transfer matrix method [5, 18], we obtain the
generating functions for the sequences hnc
mand hc
m, thereby proving that they both satisfy some linear
homogeneous recurrence relations with constant coefficients.
For each fixed m, these two generating functions share the same denominator, hence the same recur-
rence relation. We used Pascal programs and Mathematica 6 to carry out the computation. Our results
agree with those reported in [2, 11], which used a different approach. The computational data from
m= 2,4,6,8,10 suggest that hnc
m(n)and hc
m(n)have the same number of digits and start with the same
sequence of digits. For example,
hnc
10 (100) = 106189661997982901262641694866260787081353490654045349773784
008483411988691035247114502475722767402987233190282387756909
3701143503070291097245473763298031619982266082,
hc
10(100) = 106189661997133629777153967627991207437193145571362259096752
805056007992463634686046052605540587643324294617040045670714
1143497346647742593316608877569233239238111440.
Both numbers have 166 digits, and their first 12 digits are identical. Why is this happening?
2 Preliminaries
The graph Cm×Pn+1 can be drawn on a cylindrical surface in such a way that no edges cross each other,
see Figure 1. There are mn squares (4-cycles) called windows. Label the vertices (i, j )and the windows
wi,j , where 1≤i≤m,1≤j≤n+ 1 for vertices, and 1≤j≤nfor windows, as shown in Figure 1.
Construct a window lattice graph Wm,n with vertices representing the windows of Cm×Pn+1, and two
vertices are adjacent if and only if their corresponding windows in Cm×Pn+1 share a common edge. It
should be clear that Wm,n is isomorphic to Cm×Pn.
We distinguish two types of closed Jordan curves on a cylindrical surface: those that divide the surface
into two infinite regions (image the cylinder being extended indefinitely in both directions to the left and
to the right), see the curve Knc in Figure 2, and those that divide the surface into one finite and one infinite
region, see the curve Kcin Figure 2. The first type (non-contractible HC) wraps around the cylindrical
surface, hence divides the cylindrical surface into the left half and the right half, it resembles a bracelet
around an arm. The second type (contractible HC) encloses a finite region and leaves an infinite region on
the outside. One could imagine it being pasted onto the cylindrical surface.
We abbreviate these two types of Hamiltonian cycles as HCnc and HCc, respectively. We use the
following convention to name the two regions separated by a HC:

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 221
Fig. 1: The labeled graph Cm×Pn+1 and its windows.
Fig. 2: Two types of closed Jordan curves on a cylindrical surface.
•For a non-contractible HC: all edges that connect two adjacent vertices from {(i, 1) |1≤i≤m},
but do not lie on the HC, belong to the same region. We call this region (on the left of the HC) the
zero region, and the other region (on the right of the HC) the positive region.
•For a contractible HC: the windows within the bounded region are marked with 1s, hence the
bounded region is the positive region, which makes the exterior unbounded region the zero region.
Alternatively, the orientation of the HC is chosen such that the zero region is always on our left as we
traverse through the HC (see Figure 3). For HCcthis orientation is in the clockwise direction.
We use hnc
m(n)and hc
m(n)to indicate the number of HCncs and HCcs. Their respective generating
functions are written as Hnc
m(x)and Hc
m(x). Using a standard parity argument (likes the one used on a
checkerboard), it is easy to tell which thin grid cylinders have a Hamiltonian cycle.
Theorem 2.1 For m≥2and n≥1, we have hnc
m(n) = 0 if and only if both mand nare odd, and
hc
m(n) = 0 if and only if mis odd and nis even.
Proof: It is straightforward to construct a HCnc for even mor even n, and a HCcfor even mor odd n
(see Figure 4). It remains to establish the condition under which no HC exists.
Consider the “vertical” edges joining vertices (m, i)to (1, i)for 1≤i≤n+ 1, see Figure 3. Any HC
may contain some of these vertical edges, and the number of such edges is odd for a HCnc, and even for
a HCc.
As we travel along a non-contractible Hamiltonian cycle, the number of steps “to the left” and “to the
right” must be equal, while the difference between the “up” and “down” steps is m. Since the HC contains

222 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
Fig. 3: Two types of Hamiltonian cycles.
m(n+ 1) edges, we deduce that m(n+ 1) ≡m(mod 2). Thus, a HCnc does not exist if both mand n
are odd.
Similarly, if there exists a contractible Hamiltonian cycle, then m(n+ 1) must be even, because there
is an equal number of left and right steps, and an equal number of up and down steps. Hence, there is no
HCcif mis odd and nis even. 2
Hamiltonicity of a graph has both a local (every vertex is visited exactly once) and a global (the sub-
graph is connected) aspect. For a HCnc , the windows belonging to any one of the two regions induce a
forest in the window lattice graph Wm,n. We call the trees in these forests zero trees (abbreviated ZTs) or
positive trees (abbreviated PTs) depending on which region they belong to. Accordingly, their respective
windows are called zero windows or positive windows. Every zero tree contains exactly one window on
the first column of Wm,n from the set {wi,1|1≤i≤m}called the left root, and every positive tree
contains exactly one window on the last column of Wm,n from the set {wi,n |1≤i≤m}called the
right root. For example, the HCnc in Figure 3 has three zero trees with left roots w1,1, w3,1, and w7,1
(striped), and two positive trees with right roots w7,10, and w10,10 (striped).
For a HCc, the interior windows (they are marked with 1s in the HCcin Figure 3) form a tree in Wm,n,
but the exterior windows form a forest of exterior trees (abbreviated ETs). Note that only one ET from
this forest contains exactly one window on the first column of Wm,n (the left root), and also exactly one
window on the last column of Wm,n (the right root). We call this ET the split tree of the HC. Any ET
different from the split tree contains either exactly one left root or exactly one right root, but not both.
For example, the HCcin Figure 3 has a split tree with the left root w1,1and the right root w3,10, one ET
with the left root w7,1, and one ET with the right root w9,10. For the purpose of this study, interior tree

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 223
Fig. 4: The construction of the two types of Hamiltonian cycles.
and exterior trees are also called positive tree and zero trees, and their windows are labeled by 1 and 0,
respectively.
We need a few additional definitions to facilitate our discussion.
Definition 1 Given a nonnegative integer word d1d2. . . dm, its support is defined as the binary word
¯
d1¯
d2. . . ¯
dm, where
¯
di=1if di>0,
0if di= 0.
The support of a nonnegative integer matrix [di,j ]is defined in a similar manner.
Definition 2 The factor uof a word vis called a b-factor if it is a block of consecutive letters all of which
equal to b. A b-factor of vis said to be maximal if it is not a proper factor of another b-factor of v.
The approach described in the next section allows us to simultaneously analyze both types of Hamilto-
nian cycles.
3 First Characterization of HC
We associate with each Hamiltonian cycle of Cm×Pn+1 , for both types, a binary matrix [ai,j ]m×n,
denoted Anc for HCnc, and Acfor HCc, according to
ai,j =n1if wi,j is a positive window,
0otherwise.

224 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
This matrix satisfies the following necessary conditions which are easy to verify (we adopt the convention
that am+1,j =a1,j , for 1≤j≤n).
Theorem 3.1 The matrix [ai,j]m×nsatisfies the following conditions:
[A1] (First column condition): The cyclic word a1,1a2,1. . . am,1∈ {0,1}mhas at least one 0and does
not contain the factor 00.
[A2] (Adjacency condition): For each iand jwith 1≤i≤mand 1≤j≤n−1,
(ai,j , ai+1,j , ai,j+1 , ai+1,j+1 )/∈ {(1,1,1,1),(0,0,0,0),(0,1,1,0),(1,0,0,1)}.
[A3] (Root condition): Each connected component of the subgraph of Wm,n induced by the 1-windows
has a tree structure, and
•For HCnc , every positive tree has exactly one square from the last column of Wm,n.
•For HCc, there is exactly one positive tree.
[A4] (Last column condition): The cyclic word a1,na2,n . . . am,n ∈ {0,1}mhas
•For HCnc , at least one 1, and does not contain the factor 11.
•For HCc, at least one 0, and does not contain the factor 00.
It is clear that every HCnc (HCc, resp.) yields exactly one matrix Anc (Acresp.) that satisfies conditions
[A1]–[A4]. The converse is also true.
Theorem 3.2 Every matrix [ai,j]m×nwith entries from {0,1}that satisfies conditions [A1]–[A4] deter-
mines a unique HCnc (or HCc) on Cm×Pn+1.
Proof: The entries in the matrix Acan be used to label the windows of Cm×Pn+1 with 0 and 1. Construct
a subgraph on Cm×Pn+1 by forming its edges as follows. Any edge neighboring a 0-window and a 1-
window is selected. For Anc, a left edge that joins the vertices (m, 1) and (1,1), or the vertices (i, 1)
and (i+ 1,1), for 1≤i≤m−1, is selected if it is adjacent to a 1-window, and a right edge that joins
(m, n + 1) to (1, n + 1) or (i, n + 1) to (i+ 1, n + 1), for 1≤i≤m−1, is selected if it is adjacent
to a 0-window. For Ac, an edge on the left or right boundary is selected if it adjacent to a 1-window. For
example, for the matrices in Figure 3, the edge between the vertices (3, n + 1) and (4, n + 1) is selected
for Anc but not for Ac.
The conditions [A1], [A2] and [A4] imply that this subgraph of Cm×Pn+1 is a 2-factor. The global
aspect of Hamiltonicity is provided by condition [A3]. The boundary of the positive region determines
the uniqueness of the HC. 2
We note that every possible first column in both Anc and Acand last column in Acis a circular binary
words of length mwith no consecutive 0’s, and is different from the word 1m. Likewise, every possible
last column in Anc is a circular binary words of length mwith no consecutive 1’s, and is different from
the word 0m. It is well-known that the number of such binary words is Lm−1, where Lmis the mth
Lucas numbers with L0= 2,L1= 1, and Lk+1 =Lk+1 +Lkfor k≥0. See, for example, [1].

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 225
4 Second Characterization of HC
In this section, we propose an alternate characterization of the HCs on Cm×Pn+1. Although it is more
complicated, it leads to an effective way to compute the generating functions Hnc
m(x)and Hc
m(x). In the
following discussion, Adenotes either Anc or Ac.
Definition 3 Given a fixed positive integer k, two windows wi,l and wj,s that satisfy ai,l =aj,s = 1
(from either Anc or Ac) and l, s ≤kare said to be k-SIST (surely in the same tree looking from the
k-th column) if and only if they belong to the same component in the subgraph of Wm,n induced by
{wp,t |ap,t = 1 and t≤k}.
For fixed k, being k-SIST is an equivalence relation on the set {wi,k |ai,k = 1 and 1≤i≤m}and
it has at most bm/2cequivalence classes. It is possible that two different classes eventually belong to the
same positive tree of a Hamiltonian cycle on the entire cylindrical surface of Cm×Pn+1 . In other words,
two windows that are not k-SIST could become `-SIST for some integer `>k. However, we cannot tell
whether it is true just from the first kcolumns of the matrix A.
Let C+={2,3,...,bm/2c+ 1}. For any HCnc or HCc, we associate to the matrix Anc or Acfrom
the first characterization a second matrix [bi,j ]m×n, denoted Bnc or Bc, where bi,j ∈C+∪ {0}, in the
following way (see Figure 5). For each j:
(a) If ai,j = 0, then bi,j = 0.
(b) Partition the positive windows in the jth column into j-SIST equivalence classes, label all the
windows within each equivalence class 2, 3, .. . , according to the order in which the equivalence
classes first appear within the jth column, from top to bottom.
Fig. 5: The labeling of the windows of a HCnc on C10 ×P11, and a HCcon C10 ×P11 .
Theorem 4.1 The matrix B= [bi,j]m×n(either Bnc or Bc) satisfies the following properties (we adopt
the convention bm+1,j =b1,j, and b0,j =bm,j , for 1≤j≤n):

226 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
[B1] The first column b1,1b2,1. . . bm,1is either
02d103d204d3. . . 0(p+ 1)dp, p +
p
X
i=1
di=m,
or
2d103d204d3. . . 0(p+ 1)dp02(m−p−d1−d2−...−dp), p +
p
X
i=1
di≤m,
where p≥1is the number of 0s and di>0for 1≤i≤p.
[B2] The support of the matrix B, that is, the matrix [ai,j]m×n, satisfies the adjacency condition [A2].
[B3] For 1≤k≤n, the kth column of the matrix Bsatisfies these conditions:
(a) If bi,k >0, where 1≤i≤m, then bi−1,k , bi+1,k ∈ {bi,k ,0}.
(b) If bp1,k, bp2,k , . . . , bpl,k, where l≤ bm/2c, and p1< p2<··· < pl, are the first appearance
of the elements from C+in the kth column, then bpi,k =i+ 1.
(c) If k≥2,1≤i, j ≤m,i6=j,bi,k−1=bj,k−1, and ai,k =aj,k =ai,k−1=aj,k−1= 1, then
bi,k =bj,k.
(d) If k≥2,1≤i, j ≤m,i6=j,bi,k−1=bj,k−1,bi,k =bj,k =b, and ai,k−1=ai,k = 1, then
the kth column does not contain any b-factor that contains both bi,k and bj,k.
(e) If k≥2and if vand uare two different maximal nonzero b-factors in the kth column, then
there is exactly one sequence v=v1, v2, . . . , vp=uof p > 1different maximal b-factors
in the kth column with the property that for every iwith 1≤i≤p−1, in the (k−1)th
column, there exists exactly one letter bji,k−1with aji,k−1=aji,k for which bji,k ∈vi, and
there exists exactly one letter bsi+1 ,k−1with asi+1 ,k−1=asi+1,k for which bsi+1 ,k ∈vi+1 and
bji,k−1=bsi+1,k−1; and ji6=sifor 1< i < p (see Figure 6).
(f) For k≥2and for each number b∈C+that appears in the (k−1)th column, there must exist
an integer i, where 1≤i≤m, for which bi,k−1=band bi,k >0.
(g) Every column has both positive and zero entries.
[B4] The last column b1,nb2,n . . . bm,n is
•For HCnc ,
0d120d230d3. . . p0dp(p+ 1)0m−p−d1−d2−···−dp, p +
p
X
i=1
di=m,
or
20d130d240d3. . . (p+ 1)0dp, p +
p
X
i=1
di≤m,
where p≥1is the number of positive integers and di>0for 1≤i≤p.

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 227
•For HCc,
2d102d202d3. . . 02dp02m−p−d1−d2−···−dp, p +
p
X
i=1
di=m,
or
02d102d202d3. . . 02dp, p +
p
X
i=1
di≤m,
where p≥1is the number of 0s and di>0for 1≤i≤p.
Proof: First, a few remarks.
•[B1] and [B2] follow from the definition of the matrix B.
•[B3a]: Two windows belonging to the same equivalence class must be associated with the same
number.
•[B3b]: This follows from the definition of the matrix B.
•[B3c]: If wi,k−1and wj,k−1are (k−1)-SIST, and if the windows wi,k,wj,k,wi,k−1and wj,k−1
are from the positive region, then the windows wi,k and wj,k must be k-SIST.
•[B3d]: If the opposite is true, we would obtain a cycle in a positive tree, which is impossible.
•[B3e]: If we can conclude by knowing the first kcolumns that vand uare in the same tree, then
there is exactly one path from vto uin their positive tree via some windows from the previous
column, that is, the (k−1)th column.
•[B3f]: Every positive tree must “reach” the last column.
•[B3g]: For a HCnc, the unique path in Wm,n starting in a positive window from the first column
and finishing in the last column must cross every column. For a HCc, the unique split tree must
cross every column as well. Furthermore, the occurrence of a column with no zero window would
imply that the corresponding subgraph in Cm×Pn+1 is not connected, which is impossible.
•[B4]: This follows from the definition of the matrix B.
Based on these remarks, it is not difficult to verify the properties listed in the theorem. 2
Theorem 4.2 Every integer matrix B= [bi,j ]m×nwith entries from C+∪ {0}satisfying properties
[B1]–[B4] determines a unique HC on Cm×Pn+1.
Proof: It suffices to show that the support of B(which could be either Bnc or Bc) satisfies conditions
[A1]–[A4] in Theorem 3.1. It is clear that properties [B1], [B2] and [B4] imply conditions [A1], [A2]
and [A4], respectively. Properties [B3d] and [B3e] yield the forest structure for the subgraph of Wm,n
induced by positive windows (since no cycle can occur). The properties [B3c], [B3f] and [B4] for Bnc
assert that every positive tree in Wm,n has exactly one right root. For Bc, the property [B3f] implies that
for every positive window there exists a path starting from this window and finishing in the last column
of Wm,n, and the property [B4] guarantees that the subgraph of Wm,n induced by the positive windows
is connected. 2

228 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
Fig. 6: The property [B3e].
5 Technique for Enumerating Hamiltonian Cycles
For each integer m≥2, we construct a digraph Dmin the following manner. The set of vertices V(Dm)
consists of all possible columns in the matrix B. Hence, V(Dm)consists of integer words d1d2. . . dm
from the alphabet C+∪ {0}. A directed line joins the vertex vto the vertex u, where v, u ∈V(Dm),
if and only if the vertex v(as an integer word b1,k−1b2,k−1. . . bm,k−1) might be the previous column
for the vertex u(as a word b1,kb2,k . . . bm,k ). Consequently, these two words satisfy conditions [B2] and
[B3]. The subset of V(Dm)that consists of all possible first columns in the matrix B(condition [B1])
is represented by Fm. The subset of V(Dm)consisting of all possible last columns in the matrix B
(condition [B4]) is denoted Lnc
mor Lc
mdepending on whether the HC is non-contractible or contractible.
The problem of enumerating HCnc or HCcon Cm×Pn+1 now becomes the problem of enumerating
oriented walks of the length n−1in the digraph Dmwith the initial vertices in the set Fm, and the final
vertices in set Lnc
mor Lc
m. We note that Faase [7] used a similar method to enumerate spanning subgraphs
of G×Pnthat meet certain conditions.
Because of the rotational symmetry and reflection symmetry of Cm×Pn, we can further simplify the
digraph Dmby identifying some of its vertices, hence reducing its adjacency (transfer) matrix Tmto a
smaller size. By doing so, we obtain the multidigraph D∗
minstead of Dmwith transfer matrix T∗
m.

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 229
The computation of the generating functions
Hnc
m(x) =
∞
X
n≥0
hnc
m(n+ 1)xnand Hc
m(x) =
∞
X
n≥0
hc
m(n+ 1)xn
is rather routine (see Theorem 4.7.2 in [18]). It is obvious that
Hm(x) =
∞
X
n≥0
hm(n+ 1)xn=Hnc
m(x) + Hc
m(x).(1)
These generating functions are rational functions. Their denominators are determined by the characteristic
polynomials of the adjacency matrices. Table 1 displays, for 3≤m≤10, the numbers of vertices in Fm,
Dmand D∗
m, as well as the degrees of the denominators in these generating functions, which determine
the orders of the recurrence relations for hnc
mand hc
m.
We find an interesting upper bound of |V(Dm|. A column in the matrix [bi,j ]m×ncan be viewed as a
word. Let its maximal nonzero b-factors, in the order of their appearance, be p1-factor, p2-factor, . . . , pk-
factor. Call p1p2. . . pkapositive truncated word. For example, the positive truncated words correspond
to the 2nd and 6th columns of Bcin Figure 5 are 22233 and 2322, respectively. Every truncated word v
has two properties:
•If a letter s≥3appears in v, then, in accordance with the property [B3b], each number from
{2, . . . , s −1}must have appeared at least once before it. In other words, if we remove the dupli-
cated letters, the remaining letters will form the word 234. . . .
•If abab is a subsequence of the word v, then a=b(because of the properties [B3e] and [B1]).
A word over the alphabet {2, . . . , k +1}that possesses the above-mentioned properties is called a color
word. The number of color words of length kis the Catalan number Ck=1
k+1 2k
k, see [3, 16]. Using a
relation between Catalan and Motzkin numbers described in [6], we obtain the following corollary.
Corollary 5.1 An upper bound on the number of vertices of digraph Dmis
|V(Dm)| ≤ 2
bm/2c
X
k=1 m
2kCk= 2(Mm−1),
where Cmis the mth Catalan number and Mmis mth Motzkin number.
In light of Corollay 5.1, we would like to remark that we could use Motzkin words to encode the
columns. See, for example, [19].
6 Computational Results
Based on the discussion in the previous section, we use Pascal programs to compute the adjacency matri-
ces of the multidigraphs D∗
m, from which we obtain Hnc
m(x)and Hc
m(x). The results are summarized in
Table 1. Notice that the numbers |V(Dm)|and 2(Mm−1) are equal when mis odd.

230 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
m3 4 5 6 7 8 9 10
|Fm|=Lm−13 6 10 17 28 46 75 122
2(Mm−1) 6 16 40 100 252 644 1668 4374
|V(Dm)|6 12 40 64 252 364 1668 2234
|V(D∗
m)|2 4 8 14 30 44 128 172
deg den. Hm(x)1 2 3 7 12 20 51 74
deg den. Km(x)1 2 3 6 12 20 51 67
deg den. Hnc
m(x),Hc
m(x)2 4 6 13 24 40 102 141
Tab. 1: The computational results from Pascal programs.
Since Hnc
m(x)and Hc
m(x)are derived from the same transfer matrix, their denominators are identical.
After adding the two rational functions to form Hm(x), the new denominator may have a lesser degree.
In fact, numerical data reveal that the degree is reduced by roughly one-half, see Table 1.
Upon further examination of the factorization of the denominator, we conclude that a better way to
study them is to introduce the function
Km(x) = Hc
m(x)− Hnc
m(x),(2)
such that, together with (1),
Hnc
m(x) = 1
2(Hm(x)− Km(x)) ,(3)
Hc
m(x) = 1
2(Hm(x) + Km(x)) .(4)
Since both Hm(x)and Km(x)are rational functions, we can express them as
Hm(x) = Hm(x) + pm(x)
qm(x)and Km(x) = Km(x) + rm(x)
sm(x),
for some polynomials Hm(x),Km(x),pm(x),qm(x),rm(x)and sm(x), such that deg(pm)<deg(qm)
and deg(rm)<deg(sm).
The denominator qm(x)of the generating function Hm(x)provides important information about the
numbers hm(n). Let its degree be dm. Then χm(t) = tdmqm(1/t)is the characteristic polynomial which
determines the recurrence relation that hm(n)satisfies. It has dmnonzero roots (the characteristic roots)
over C, name them λm,i so that |λm,1|≥|λm,2| ≥ ··· ≥ |λm,dm|. We can write
qm(x) =
dm
Y
i=1
(1 −λm,ix).
Note that the zeros of qm(x)are λ−1
m,i. For the sake of brevity, we shall still call λm,is the characteristic
roots of qm(x). It is a routine exercise to show that, if λm,is are simple (hence distinct) roots, then
pm(x)
qm(x)=
dm
X
i=1
αi
1−λm,ix,

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 231
so that for sufficiently large n
hm(n+ 1) =
dm
X
i=1
αiλn
m,i,
where αi=−λm,ipm(λ−1
m,i)/q0
m(λ−1
m,i). The solution is more complicated if some of the λm,i s are
repeated roots. Nonetheless, if λm,1is a simple positive root such that λm,1>|λm,2|, then
hm(n+ 1) ∼α1λn
m,1,
in which the formula for α1given above still holds. See the following sections for illustrations of our
discussion.
6.1 The Thin Grid Cylinder C2×Pn+1
We find hnc
2(n) = 2 and hc
2(n)=2, hence h2(n)=4, for all n≥1.
6.2 The Thin Grid Cylinder C3×Pn+1
Let V(D3) = {v1, v2, . . . , v6}. We obtain the following:
v1= (2,2,0)
v2= (2,0,2)
v3= (0,2,2)
v4= (0,0,2)
v5= (0,2,0)
v6= (2,0,0)
T3=








000011
000101
000110
011000
101000
110000








F3={v1, v2, v3}
Lnc
3={v4, v5, v6}
Lc
3={v1, v2, v3}
T∗
3=0 2
2 0 
hnc
3(2k−1) = 0, k ≥1
hnc
3(2) = 6
hnc
3(4) = 24
hnc
3(6) = 96
hnc
3(8) = 384
hc
3(2k)=0, k ≥1
hc
3(1) = 3
hc
3(3) = 12
hc
3(5) = 48
hc
3(7) = 192
The characteristic polynomial of T∗
3is x2−4. Because of Cayley-Hamilton theorem, we obtain the
recurrence relations hnc
3(n) = 4hnc
3(n−2) and hc
3(n) = 4hc
3(n−2). The generating functions are
Hnc
3(x) = 6x
1−4x2=3
2(1 −2x)−3
2(1 + 2x),
Hc
3(x) = 3
1−4x2=3
2(1 −2x)+3
2(1 + 2x).
Therefore,
H3(x) = 3
1−2xand K3(x) = 3
1+2x.
The denominator of H3(x)yields the recurrenece relation
h3(n)=2h3(n−1), n ≥2.
Since 3
1−2x= 3 P∞
k=0 2kxk, we obtain the following simple formula for h3(n).

232 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
Theorem 6.1 For n≥1, the number of Hamiltonian cycles in C3×Pn+1 is
h3(n)=3·2n−1.
6.3 The Thin Grid Cylinder C4×Pn+1
Let V(D4) = {v1, v2, . . . , v12}. We obtain the following:
v1= (2,2,2,0)
v2= (2,2,0,2)
v3= (2,0,2,2)
v4= (2,0,3,0)
v5= (0,2,2,2)
v6= (0,2,0,3)
v7= (0,0,0,2)
v8= (0,0,2,0)
v9= (0,2,0,0)
v10 = (2,0,0,0)
v11 = (2,0,2,0)
v12 = (0,2,0,2)
T4=




















000000011110
000000101101
000000110110
101100000000
000000111001
010011000000
011011000000
101110000000
110011000000
111100000000
000000010110
000000101001




















F4={v1, v2, . . . , v6}
Lnc
4={v4, v6, v7, v8, v9, v10}
Lc
4={v1, v2, v3, v5, v11, v12 }
T∗
4=



0031
2100
3100
0021




hnc
4(1) = 2
hnc
4(2) = 14
hnc
4(3) = 34
hnc
4(4) = 170
hnc
4(5) = 530
hnc
4(6) = 2230
hnc
4(7) = 7714
hnc
4(8) = 30258
hnc
4(9) = 109378
hnc
4(10) = 416766
hnc
4(11) = 1534722
hnc
4(12) = 5777562
hnc
4(13) = 21441682
hc
4(1) = 4
hc
4(2) = 8
hc
4(3) = 48
hc
4(4) = 136
hc
4(5) = 612
hc
4(6) = 2032
hc
4(7) = 8192
hc
4(8) = 29104
hc
4(9) = 112164
hc
4(10) = 410040
hc
4(11) = 1550960
hc
4(12) = 5738360
hc
4(13) = 21536324
The generating functions are:
Hnc
4(x) = 2(1 + 5x−5x2+x3)
(1 −4x+x2)(1 + 2x−x2)=3−x
1−4x+x2−1−x
1+2x−x2,
Hc
4(x) = 4
(1 −4x+x2)(1 + 2x−x2)=3−x
1−4x+x2+1−x
1+2x−x2,
from which we obtain
H4(x) = 2(3 −x)
1−4x+x2and K4(x) = 2(1 −x)
1+2x−x2,

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 233
and the recurrence relation
h4(n) = 4h4(n−1) −h4(n−2), n ≥3.
After decomposing into partial fractions, we find
2(3 −x)
1−4x+x2=9+5√3
3·1
1−2 + √3x+9−5√3
3·1
1−2−√3x.
This leads to the next result.
Theorem 6.2 For n≥1, the number of Hamiltonian cycles in C4×Pn+1 is
h4(n) = 1
39+5√32 + √3n−1+9−5√32−√3n−1,
and h4(n)∼1
39+5√32 + √3n−1.
6.4 The Thin Grid Cylinder C5×Pn+1
We find |V(D5)|= 40,V(D∗
5) = {v1, . . . , v8}, and
v1= (2,2,2,2,0)
v2= (2,2,0,3,0)
v3= (0,0,0,0,2)
v4= (0,0,2,0,3)
v5= (2,0,2,0,0)
v6= (2,0,0,2,2)
v7= (2,0,2,2,0)
v8= (2,0,0,0,2)
T∗
5=












00403000
00020200
43000000
32000000
00000022
00000022
00302000
00020200












hnc
5(2k−1) = 0, k ≥1
hnc
5(2) = 30
hnc
5(4) = 850
hnc
5(6) = 24040
hnc
5(8) = 680040
hnc
5(10) = 19236840
hc
5(2k) = 0, k ≥1
hc
5(1) = 5
hc
5(3) = 160
hc
5(5) = 4520
hc
5(7) = 127860
hc
5(9) = 3616880
We obtain
Hnc
5(x) = 10x(x2+ 3)
1−28x2−8x4−4x6=5
1−6x+ 4x2−2x3−5
1+6x+ 4x2+ 2x3,
Hc
5(x) = 5(4x2+ 1)
1−28x2−8x4−4x6=5
1−6x+ 4x2−2x3+5
1+6x+ 4x2+ 2x3.
Hence,
H5(x) = 10
1−6x+ 4x2−2x3and K5(x) = 10
1+6x+ 4x2+ 2x3.
Due to its complexity, we will not display the explicit formula for h5(n). Numerically, λ5,1≈5.31863,
and λ5,2, λ5,3≈0.34069 ±0.50987i.

234 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
6.5 The Thin Grid Cylinder C6×Pn+1
Hnc
6(x) = 2 + 62x+ 278x2+ 4178x3+ 27710x4+ 314354x5+ 2468810x6+ 24770708x7
+ 210413420x8+ 1998760352x9+ 17601771968x10 + 163119159176x11
+ 1460403914672x12 + 13382718140000x13 + 120722781112208x14
+ 1100628776882000x15 + 9962793339446672x16
+ 90619491133658576x17 + 821568683907144752x18
+ 7464893093725073072x19 + 67726216376743239056x20 +··· ,
Hc
6(x) = 6 + 24x+ 498x2+ 2832x3+ 35964x4+ 263736x5+ 2779014x6+ 22869384x7
+ 222067212x8+ 1927331160x9+ 18039580560x10 + 160435712688x11
+ 1476851478768x12 + 13281906604320x13 + 121340682078768x14
+ 1096841495972016x15 + 9986006600900208x16 + 90477210822238320x17
+ 822440758133272176x18 + 7459547916670820976x19
+ 67758978401907276048x20 +··· ,
H6(x) = 2(4 + 7x+x2−27x3−26x4−20x5−3x6)
1−9x−10x3+ 28x4+ 36x5+ 32x6+ 12x7,
K6(x) = 2(2 −11x+ 14x2−11x3−x4+x5)
1+4x−10x2+ 16x3−16x4+ 4x5+ 4x6.
The denominator q6(x)has seven simple roots, three real and four complex, and λ6,1≈9.07807.
6.6 The Thin Grid Cylinder C7×Pn+1
We find H7(x) = p7(x)/q7(x), and K7(x) = r7(x)/s7(x), where
Hnc
7(x) = 126x+ 18452x3+ 2861964x5+ 444486280x7+ 69048910000x9
+ 10726732430288x11 + 1666401898058352x13 + 258876295158900832x15
+ 40216553455854426560x17 + 6247660438430706481984x19 +··· ,
Hc
7(x) = 7 + 1484x2+ 229698x4+ 35663964x6+ 5539931796x8+ 860620499760x10
+ 133697577587000x12 + 20769976722986288x14 + 3226625529605854320x16
+ 501257787787122948736x18 + 77870632467402116097056x20 +··· ,
p7(x) = 7(1 + 6x−22x2−120x3−178x4+ 72x5+ 580x6+ 616x7+ 264x8+ 72x9+ 16x10),
q7(x)=1−12x−18x2+ 112x3+ 440x4+ 772x5+ 196x6
−2064x7−3724x8−2040x9−496x10 −128x11 + 16x12,
and r7(x) = p7(−x), and s7(x) = q7(−x).

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 235
6.7 The Thin Grid Cylinder C8×Pn+1
Again, we have H8(x) = K8(x)=0,
Hnc
8(x) = 2 + 254x+ 1794x2+ 82138x3+ 1012930x4+ 30717374x5+ 481369234x6
+ 12070287370x7+ 214585144402x8+ 4886085696654x9
+ 92880601782338x10 + 2011688161424970x11 + 39622707294281746x12
+ 836009740378418718x13 + 16778455639135020178x14 +··· ,
Hc
8(x) = 8 + 64x+ 4320x2+ 44288x3+ 1575288x4+ 22337664x5+ 605992784x6
+ 10215798448x7+ 242178636928x8+ 4475508186384x9
+ 98989761676840x10 + 1920787160180224x11 + 40975264449253872x12
+ 815884428197037360x13 + 17077909293201385648x14 +··· ,
p8(x) = 2(5 + 44x−430x2+ 33x3+ 93x4+ 1471x5+ 4596x6+ 6807x7
+ 8263x8+ 2751x9−2482x10 −5126x11 −4711x12 −2094x13
−1406x14 + 450x15 + 580x16 −132x17 + 32x18 + 40x19),
q8(x)=1−23x+ 34x2+ 345x3+ 218x4−22x5−2919x6−5041x7
−8806x8−11998x9−5873x10 + 1318x11 + 4467x12 + 11373x13
+ 3848x14 −584x15 + 1018x16 −928x17 + 84x18 + 72x19 −40x20,
r8(x) = 2(3 −80x+ 476x2−1143x3+ 303x4+ 4917x5−8670x6−2291x7
+ 19477x8−13315x9−16780x10 + 19224x11 + 6103x12 −9974x13
−1352x14 + 3926x15 −1796x16 + 644x17 −168x18 + 16x19),
s8(x) = 1 + 5x−104x2+ 529x3−1548x4+ 1830x5+ 3915x6−13527x7
+ 7182x8+ 20914x9−31027x10 −9214x11 + 35037x12 + 1205x13
−19590x14 + 890x15 + 5770x16 −2048x17 + 588x18 −184x19 + 16x20.
6.8 The Thin Grid Cylinder C9×Pn+1
Hnc
9(x) = 510x+ 351258x3+ 276018090x5+ 218915964618x7+ 173923080282474x9
+ 138226113213225360x11 + 109864493967924549384x13
+ 87323767337933601800838x15 + 69407973132514050824027916x17
+ 55167927811346067821770238916x19
+ 43849442381504976630009404305836x21 +··· ,
Hc
9(x) = 9 + 12348x2+ 9806292x4+ 7769376972x6+ 6169925169414x8
+ 4903042542453720x10 + 3896923927019062734x12
+ 3097380080814655131414x14 + 2461902328199084994926838x16
+ 1956807009306757665486727506x18
+ 1555340096869096304430909957438x20 +··· .

236 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
We find H9(x) = K9(x) = 0. Like the cases of m= 3,5,7, we also have r9(x) = p9(−x)and
s9(x) = q9(−x). However, since deg(p9) + 1 = deg(q9) = 51, we will not attempt to list these
polynomials in their entirety.
6.9 The Thin Grid Cylinder C10 ×Pn+1
Hnc
10 (x) = 2 + 1022x+ 10652x2+ 1505612x3+ 32718482x4+ 2701992092x5+ 79977736982x6
+ 5099841986502x7+ 179765502917052x8+ 9933064485778002x9
+ 387981888303174142x10 + 19745599426500473672x11
+ 819563054782862759352x12 + 39759941758256449144532x13
+ 1710706207634346787583712x14 + 80696804239003472593910602x15 +···
Hc
10(x) = 10 + 160x+ 34850x2+ 621720x3+ 62999960x4+ 1641664580x5+ 116791523380x6
+ 3817933082020x7+ 224360971248960x8+ 8381173203185000x9
+ 441980748032029010x10 + 17866610320162579120x11
+ 884945074721799980580x12 + 37484874131377414126080x13
+ 1789870555278304706976120x14 + 77942162101044243981212480x15 +··· .
We close by mentioning that H10 (x) = K10(x)=0,deg(p10 ) + 1 = deg(q10) = 74, and deg(r10) + 1 =
deg(s10) = 67.
7 Asymptotic Values
Let ρmbe the radius of convergence for Hm(x). The coefficients of Hm(x)are non-negative, Pring-
sheim’s Theorem (see, for example, [8]) states that it has a singularity at x=ρm. Since we assume that
qm(x) = Qdm
i=1(1 −λm,i x), where |λm,1|≥|λm,2|≥···≥|λm,dm| 6= 0, one of the characteristic roots
with the largest moduli must be real, positive, and equal to 1/ρm. We may assume it is λm,1. For brevity,
we denote it θm. If θm=λm,1>|λm,2|, then θmis the dominant root, and
hm(n+ 1) ∼amθn
m,
where am=−θmpm(θ−1
m)/q0
m(θ−1
m). Do we always have |λm,1|>|λm,2|? The fact that the transfer
matrix T∗
mis nonnegative points to the Perron-Frobenius theorem for an answer.
Let Mbe a nonnegative square matrix. We say that Mis irreducible if, for every iand j, there exists a
positive integer k=k(i, j)such that (Mk)ij >0. This is equivalent to saying that the multidigraph GM
with adjacency matrix Mis strongly connected. The matrix Mis said to be primitive if Aγ>0for some
positive integer γ. For example, the matrix
T∗
3=0 2
2 0 
is irreducible but not primitive, and T∗
4is primitive because T∗
42>0. The period or order of cyclicity of
M, labeled by g, can be defined as the greatest common divisor of the lengths of the directed cycles in
GM[12]. From the Perron-Frobenius theory (see, for example, [12]), gis the number of eigenvalues of

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 237
Mhaving the largest modulus. In particular, a primitive matrix is an irreducible nonnegative matrix with
g= 1, it has exactly one dominant characteristic root. We find that, for 3≤m≤10, the transfer matrix
T∗
mis irreducible, but it is primitive only when mis even. Accordingly, we shall study the cases of odd
and even mseparately.
When mis odd, Theorem 2.1 implies that
hm(n) = hnc
m(n)if nis even,
hc
m(n)if nis odd. (5)
Hence Hnc
m(x)comes from the odd terms of Hm(x), and Hc
m(x)from the even terms. This means
Hnc
m(x) = 1
2(Hm(x)− Hm(−x)) ,
Hc
m(x) = 1
2(Hm(x) + Hm(−x)) .
Hence, Km(x) = Hm(−x)when mis odd. Since Hnc
m(x)and Hc
m(x)share the same denominator
qm(x)qm(−x), both sequences hnc
mand hc
msatisfy a linear recurrence relation of order 2dm. However,
qm(x)qm(−x)is an even function, so it is a polynomial of degree dmin x2. Thus, the subsequences
of nonzero terms {hnc
m(2n)}n≥1and {hc
m(2n−1)}n≥1satisfy a linear recurrence relation of order dm.
Because of (5), it is clear that, for the nonzero terms, the asymptotic behavior of hnc
m(n)and hc
m(n)is
same as that of hm(n). More precisely, hnc
m(2n)∼hm(2n), and hc
m(2n+ 1) ∼hm(2n+ 1).
If the transfer matrix D∗
mis irreducible, then it would have gmdominant characteristic roots, where gm
denotes the period of D∗
m. The fact that qm(x)qm(−x), the denominator that Hnc
m(x)and Hc
m(x)share,
is a polynomial in x2suggests that D∗
mis a bipartite graph. If this can be confirmed, then gmmust be
even. In fact, it contains the following directed cycle of length 2: u1u2u1, where u1and u2are vertices
(written as words) in D∗
m
u1= 222 ·· · 20, u2= 000 ···02.
See Figure 4 and note that, because of the rotational symmetry, the vertex 22 ·· · 202 is identified to
22 ···220, as well 20 ···00 to 00 ···02. We conclude that gm= 2, so the two dominant characteristic
roots of D∗
mmust be ±θm. This in turn implies that θmis the sole dominant characteristic root of qm(x).
Consequently, we deduce that, for odd m,
hnc
m(2n)∼amθ2n−1
mand hc
m(2n+ 1) ∼amθ2n
m,
provided that T∗
mis irreducible, and D∗
mis a bipartite graph. Our computational data reveal that D∗
3,D∗
5,
D∗
7,D∗
9are bipartite multidigraphs.
For even m, we note that D∗
mcontains loops. For example, there is a loop around the vertex representing
the word 2030 ·· · ((m+ 2)/2)0, see Figure 4. We conclude that, if T∗mis irreducible (recall that our
computational data confirm that the matrix T∗
mis indeed irreducible for m≤10), then gm= 1. But
the dominant characteristic root can come from either Hm(x)or Km(x). Our computational data reveal
that, for m≤10, the radius of convergence for Km(x)is greater than that of Hm(x). This, together with
(3) and (4), imply that the dominant characteristic root of both Hnc(x)and Hc
m(x)comes from Hm(x).
Hence,
hnc
m(n+ 1) ∼am
2θn
mand hc
m(n+ 1) ∼am
2θn
m.
This immediately proves that hnc
m(n)∼hc
m(n)when m= 2,4,6,8,10. Is it always true when mis even?

238 Olga Bodroˇ
za-Panti ´
c, Harris Kwong, Milan Panti´
c
8 Concluding Remarks and Open Problems
Our computational data affirm that for 3≤m≤10, the denominator qm(x)has only one real positive
dominant characteristic root θm, see Table 2. Our main conjecture is:
m θmam
3 2 3
4 3.73205080756887729352744634151 5.8867513459481288225457439025
5 5.31862821775018565910968015332 5.6485507137110988135657454508
6 9.07807499686426137037316693063 9.3759765980423268475201653010
7 12.46396683154921167484924057847 9.5114780466647699643291510197
8 20.49548062885849319891140410573 14.6698889618659187804647562240
9 28.19283279845402927227773603077 15.4543604331204162432381530254
10 45.31795107579019470088202555080 22.7172562899371282508816262267
Tab. 2: The approximate values of θmand am.
Conjecture 1 For each even m≥4,
hnc
m(n+ 1) ∼hc
m(n+ 1) ∼am
2θn
m,
where am=−θmpm(θ−1
m)/q0
m(θ−1
m).
As we have discussed in the previous section, the validity this conjecture, and other related asymptotic
relations, can be completely resolved if we can settle the following open problems:
1. Is T∗
mirreducible for all m≥3? Note that this has been confirned for m≤10.
2. Is D∗
mbipartite when mis odd? Again, this has been confirmed up to m= 9.
3. Does the dominant characteristic root remain in Hmwhen mis even? This is equivalent to showing
that the radius of convergence for Km(x)is greater than that of Hm(x).
Our computational data suggest further problems for investigation:
4. Is the sequence generated by Km(x)always alternating? In other words, do we always have
hc
m(2k)< hnc
m(2k)and hc
m(2k+ 1) > hnc
m(2k+ 1)?
We close our discussion with three more interesting questions:
6. What is an appropriate combinatorial interpretation for Km(x)?
7. Can we define a labeling of the windows such that a single transfer matrix can be used to obtain
the generating function Hm(x)directly? If such a transfer matrix does exist, its characteristic
polynomial should be χm(t)mentioned in Section 6. (Recall that the matrices obtained in [2] for
graph Cm×Pnare transfer matrices for sequences hm(n), but obtained by a labeling of the vertices
of Cm×Pn. We additionally verified that they are indeed primitive for m≤12.)
8. Can we find some similar properties of sequences hm(n),hnc
m(n)and hc
m(n)for the case of thick
cylinder Pm×Cn(mis kept constant, whereas ngrows)?

A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs 239
Acknowledgments
We are indebted to the anonymous referees. Their generous and valuable suggestions helped us reshape
the exposition in Sections 6–8. We are grateful to Dragoˇ
s Cvetkovi´
c and Jasmina Teki´
c for their con-
structive comments and suggestions for improving the paper. The authors thank Bojana Panti´
c for many
valuable comments and help in the implementation of the Pascal programs.
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