Weakly Dimension-Balanced Hamiltonian Cycle on Three-Dimensional Toroidal Mesh Graph ^†

Abstract

The dimension-balanced cycle (DBC) problem is new in graph theory, with applications such as 3D stereogram reconstruction. In a graph whose edges are partitioned into k dimensions, a cycle is dimension-balanced if edge counts across dimensions differ by at most one. When such a cycle is Hamiltonian, it is called a dimension-balanced Hamiltonian cycle (DBH). Since DBHs do not always exist, a relaxed notion—the weakly dimension-balanced Hamiltonian (WDBH) cycle—was considered, allowing a difference of up to three. We prove that WDBH always exists in any 3-dimensional toroidal mesh graph T_m,n,r for all positive integers m, n, and r.

1. Introduction

The topology of interconnection networks is modeled using graphs, where vertices represent processors and edges denote the communication links between them. In this study, we follow the standard notation and definitions introduced in Ref. [1].

In a simple graph G = (V, E), a Hamiltonian path is a path that visits each vertex exactly once; a Hamiltonian cycle is a cycle that visits each vertex exactly once [2,3,4,5,6,7]. Given a graph G whose edge set is partitioned into k dimensions, denoted as $\tilde{E}$ = {E₁(G), E₂(G), …, E_k(G)}. A cycle C in G is called a dimension-balanced cycle (DBC for short) with respect to $\tilde{E}$ if for any pair i, j ∈ {1, 2, …, k}, the number of edges in each dimension satisfies ||E_i(C)| − |E_j(C)|| ≤ 1, where E_i(C) = E(C) ∩ E_i(G) for i ∈ {1, 2, …, k}. If such a cycle is also Hamiltonian, it is referred to as a dimension-balanced Hamiltonian cycle (DBH). If the condition is relaxed into ||E_i(C)| − |E_j(C)|| ≤ 3, then the cycle is called a weakly dimension-balanced (Hamiltonian) cycle (WDBC (WDBH) for short) with respect to $\tilde{E}$.

The DBH problem has attracted increasing attention in recent years. The DBH problem was developed to address the quality issues in the construction of 3D surface scanning [8,9,10]. Notable research efforts have examined such cycles in various graph classes, including hypercube and toroidal meshes [11,12]. In 2012, Peng [11] successfully constructed DBHs for two-dimensional toroidal graphs T_m,n, and in 2018, Wu [13] extended this to the weaker version known as the weak DBH (WDBH).

We examined the class of graphs denoted by T_m,n,r, known as 3-dimensional toroidal mesh graphs (3D toroidal mesh graphs). The vertex set of such a graph is defined as V(T_m,n,r) = {(x, y, z) | 0 ≤ x < m − 1, 0 ≤ y < n − 1, 0 ≤ z < r − 1}; the edge set is given by E(T_m,n,r) = {(x₁, y₁, z₁)(x₂, y₂, z₂) | (x₁ = x₂, y₁ = y₂, |z₁ − z₂| ≡ 1 (mod r)) or (x₁ = x₂, z₁ = z₂, |y₁ − y₂| ≡ 1 (mod n)) or (y₁ = y₂, z₁ = z₂, |x₁ − x₂| ≡ 1 (mod m))}. Let the partition of E(T_m,n,r) be $\tilde{E}$ = {E₁, E₂, E₃}, where E₁ = {(x₁, y₁, z₁)(x₂, y₂, z₂) | y₁ = y₂, z₁ = z₂, |x₁ − x₂| ≡ 1 (mod m)}, E₂ = {(x₁, y₁, z₁)(x₂, y₂, z₂) | x₁ = x₂, z₁ = z₂, |y₁ − y₂| ≡ 1 (mod n)}, and E₃ = {(x₁, y₁, z₁)(x₂, y₂, z₂) | x₁ = x₂, y₁ = y₂, |z₁ − z₂| ≡ 1 (mod r)}. For convenience in further discussions, we define the following three sets of special cross-connections: cross¹ = {(0, j, k), (m − 1, j, k) | 0 ≤ j < n, 0 ≤ k < r}, cross² = {(i, 0, k), (i, n − 1, k) | 0 ≤ i < m, 0 ≤ k < r}, and cross³ = {(i, j, 0), (i, j, r − 1) |0 ≤ i < m, 0 ≤ j < n}. An illustration of the structure of T₅,₄,₃ is shown in Figure 1. Throughout this paper, edges of different dimensions in the figures are represented using three colors: red, blue, and green.

Building on the previous foundations, Juan et al. [14] investigated the DBH problem in 3D toroidal mesh graphs with respect to $\tilde{E}$ in 2019, and the rest of this problem has been solved recently [15]. However, DBHs do not always exist in arbitrary 3D toroidal graphs. Consequently, when a DBH is unattainable, the weaker form, WDBH, becomes a useful generalization. This paper aims to explore the existence conditions of WDBH cycles in 3D toroidal mesh graphs T_m,n,r with respect to $\tilde{E}$, under the constraint that m, n, r ≥ 3.

2. Background Knowledge

We analyze the properties of the 3D toroidal mesh graph T_m,n,r, based on the values of m, n, and r mod 6 at first. Then, we present important results.

Let a = m mod 6, b = n mod 6, and c = r mod 6. Considering the graph’s inherent symmetry, 56 cases need to be discussed when m, n, r ≥ 3. In 2015, Peng established an important result regarding d-dimensional toroidal mesh graphs of the form C_l1 × C_l2 ×… × C_ld, where each C_li is a cycle graph of length l_i, with 1 ≤ i ≤ d. Define the function ρ(l₁, l₂, …, l_d) as the number of odd integers in the set {l₁, l₂, …, l_d} [11]. The function Ψ(l₁, l₂, …, l_d) is defined as follows: if ⌊(l₁l₂…l_d)/d⌋ is even, then Ψ(l₁, l₂, …, l_d) = (l₁l₂…l_d) mod d; else Ψ(l₁, l₂, …, l_d) = d − ((l₁l₂…l_d) mod d). The results for the DBH problem on the 3D toroidal mesh graph T_m,n,r are as follows.

Theorem 1 

([14]). If any of m, n, or r is divisible by 3, then a DBH exists in T_m,n,r with respect to $\tilde{E}$.

Theorem 2 

([15]). If m, n, r mod 3 ≠ 0 and Ψ(m, n, r) ≤ ρ(m, n, r), then a DBH exists in T_m,n,r with respect to $\tilde{E}$.

Theorem 3 

([14]). If Ψ(m, n, r) > ρ(m, n, r), there does not exist DBH in T_m,n,r with respect to $\tilde{E}$.

By definition, every DBH is also a WDBH.

Table 1. Cases with no DBH in T_m,n,r for m, n, r ≥ 3.

Cases	a	b	c	Cases	a	b	c
1	4	4	4	6	4	8	7
2	4	4	5	7	4	8	8
3	4	4	7	8	8	8	5
4	4	4	8	9	8	8	7
5	4	8	5	10	8	8	8

summarizes ten specific cases of toroidal mesh graphs that have been shown not to admit DBHs. Since we are concerned with the cases where m, n, r ≥ 3, the range of a, b, and c has been restricted to {3, 4, 5, 6, 7, 8}. These cases are of particular interest, as it remains to be determined whether a WDBH exists in each of them. This issue will be examined in detail in the following section.

We present two lemmas that demonstrate how two cycles can be connected in this paper. Figure 2 and Figure 3 illustrate Lemma 1 and Lemma 2.

Lemma 1 

([15]). Let there be two cycles: C₁ = 〈a₁, …, a₂, a₃, …, a₄, a₁〉, C₂ = 〈b₁, …, b₂, b₃, …, b₄, b₁〉 By removing edges a₄a₁, a₂a₃, b₄b₁, b₂b₃ and adding new edges a a₁b₄, b₃a₃, a₄b₁, b₂a₂, we form a new cycle C₃ defined as C₃ = 〈a₁, b₄, (b₄, b₃)- from C₂⁻¹, b₃, a₃, (a₃, a₄)- from C₁, a₄, b₁, (b₁, b₂)- from C₂, b₂, a₂, (a₂, a₁)- from C₁⁻¹, a₁〉.

Lemma 2 

([15]). Building on Lemma 1, assume T₁ and T₂ are Hamiltonian cycles in the toroidal mesh graphs G₁ = T_m,n,r1 and G₂ = T_m,n,r2, respectively. Suppose the edges a₂a₃, a₄a₁ lie in the cross³ direction of G₁, and the edges b₂b₃, b₂b₃ lie in the cross³ direction of G₂. Then, the added edges a₁b₄, b₃a₂ will also lie in the cross³ direction of the combined graph T_m,n,r1+r2.

3. Result and Discussion

Using the modular values of n, m, and r modulo 6, the structure T_m,n,r is classified into ten distinct cases, as shown in Table 1. This section constructs the WDBH for each of the cases presented in Table 1, using the model illustrated in Figure 4. At first, we show the main idea of how to construct the WDBH for every case.

If we find one WDBH (A) and seven Hamiltonian cycles (B through H), then connecting them sequentially according to the construction method shown in Figure 4, we can cover all the various cases in

Table 2. Ten cases discussed in this study.

Case	A (a,b,c)	B (a,b,6)	C (6,b,c)	D (6,b,6)	E (a,6,c)	F (a,6,6)	G (6,6,c)	H (6,6,6)
1	B1 (4,4,4)	B2 (4,4,6)	B3 (6,4,4)	B4 (6,4,6)	B5 (4,6,4)	B6 (4,6,6)	B7 (6,6,4)	B8 (6,6,6)
2	B9 (4,4,5)	B2	B10 (6,4,5)	B4	B11 (4,6,5)	B6	B12 (6,6,5)	B8
3	B13 (4,4,7)	B2	B14 (6,4,7)	B4	B15 (4,6,7)	B6	B16 (6,6,7)	B8
4	B17 (4,4,8)	B2	B18 (6,4,8)	B4	B19 (4,6,8)	B6	B20 (6,6,8)	B8
5	B21 (4,8,5)	B22 (4,8,6)	B23 (6,8,5)	B24 (6,8,6)	B11	B6	B12	B8
6	B25 (4,8,7)	B22	B26 (6,8,7)	B24	B15	B6	B16	B8
7	B27 (4,8,8)	B22	B28 (6,8,8)	B24	B19	B6	B20	B8
8	B29 (8,8,5)	B30 (8,8,6)	B23	B24	B31 (8,6,5)	B32 (b,6,6)	B12	B8
9	B33 (8,8,7)	B30	B26	B24	B34 (8,6,7)	B32	B16	B8
10	B35 (8,8,8)	B30	B28	B24	B36 (8,6,8)	B32	B20	B8

Red bold text denotes values other than 6 and indicates the main discrepancies among the different cases.

.

During the extension along the first or second dimension, we remove one edge from each of the two Hamiltonian cycles in the same dimension and reconnect them using two new edges, also in the same dimension. As a result, the resulting graph remains a Hamiltonian cycle, and the number of edges in each dimension can simply be summed directly. When extending along the third dimension, a different construction is required. Specifically, we must remove two edges from each of the original Hamiltonian cycles in the third dimension and reconnect them using four edges in the third dimension. To ensure that the resulting structure remains a Hamiltonian cycle, each individual Hamiltonian cycle must satisfy the conditions specified in Lemma 2. According to Lemma 2, this guarantees that the merged graph is indeed Hamiltonian (still one cycle). In terms of edge count, the number of edges in each dimension can simply be summed directly, too.

We present the main theorem of this study as follows.

Theorem 4. 

For any m, n, or r ≥ 3, WDBH exists in T_m,n,r with respect to $\tilde{E}$.

Proof. 

From the discussion in the previous section, we observe that the validity of this theorem hinges on the ten specific cases listed in Table 1, where no DBH exists. Therefore, to establish this theorem, it is sufficient to prove that each of these ten cases admits a WDBH. Following the construction framework outlined at the beginning of this section, due to space limitations and the similarity in the proof techniques across all cases, we provide a detailed proof for Case 1 only.

Case 1. m mod 6 = 4 and n mod 6 = 4, r mod 6 = 4. Let B₁ (see Figure 5a) be a WDBH of T_4,4,4. We can see that |E₁(B₁)| = 20, |E₂(B₁)| = 22, and |E₃(B₁)| = 22. Let B₂ (see Figure 5b) be a DBH of T₄,₄,₆, where |E₁(B₂)| = |E₂(B₂)| = |E₃(B₂)| = 32. After connecting B₂ z times in the third dimension of B₁, it forms a WDBH T₁ in T_4,4,(4+6z). Note that |E₁(T₁)| = 20 + 32z, |E₂(T₁)| = 22 + 32z, and |E₃(T₁)| = 22 − 2 + 2 + (32 − 2 + 2)z = 22 + 32z. Let B₃ (see Figure 5c) be a DBH of T₆,₄,₄ with |E₁(B₃)| = |E₂(B₃)| = |E₃(B₃)| = 32, connecting a DBH B₄ of T_6,4,6 (with |E₁(B₄)| = |E₂(B₄)| = |E₃(B₄)| = 48, see Figure 5d) z times to the third dimension of B₃. It forms a DBH T₂ in T_6,4,(4+6z) with |E₁(T₂)| = |E₂(T₂)| = 32 + 48z, |E₃(T₂)| = 32 − 2 + 2 + (48 − 2 + 2)z = 32 + 48z. Then, connecting T₂ z times to the first dimension of T₁, it forms a WDBH in T_{(4+6x),4,(4+6z)}, called T₃. So, |E₁(T₃)| = 20 + 32z − 1 + 1 + x(32 + 48z − 1 + 1) = 20 + 32z + x(32 + 48z), |E₂(T₃)| = |E₃(T₃)| = 22 + 32z + x(32 + 48z).

In the next stage, we need to add a DBH of T_{(4+4x),6,(4+6z)} y times in the second dimension of T₃. Again, we give a DBH B₅ of T_4,6,4, as Figure 5e, and connect a DBH B₆ of T_4,6,6 z times to the third dimension of B₅. It forms a DBH T₄ in T_4,6,(4+6z) with |E₁(T₄)| = |E₂(T₄)| = 32 + 48z, |E₃(T₄)| = 32 − 2 + 2 + (48 − 2 + 2)z = 32 + 48z. Now, let B₇ (see Figure 5g) be a DBH of T_6,6,4, and B₈ (see Figure 5h) be a DBH of T₆,₆,₆. After connecting B₈ z times in the third dimension of B₇, it forms a DBH T₅ in T_6,6,(4+6z). Note that |E₁(T₅)| = |E₂(T₅)| = 48 + 72z, |E₃(T₅)| = 48 − 2 + 2 + (72 − 2 + 2)z = 48 + 72z. After connecting T₅ x times to the first dimension of T₄, it forms a DBH T₆ of T_{(4+6x),6,(4+6z)}. Note that |E₁(T₆)| = 32 + 48z − 1 + 1 + x(48 + 72z − 1 + 1) = 32 + 48z + x(48 + 72z), |E₂(T₆)| = |E₃(T₆)| = 32 + 48z + x(48 + 72z). Finally, after connecting T₆ y times to the second dimension of T₃, it forms a WDBH T₇ of T_{(4+6x),(4+6y),(4+6z)}. Note that |E₁(T₇)| = 20 + 32z + x(32 + 48z) + y [32 + 48z + x(48 + 72z)], |E₂(T₇)|= 20 + 32z + x(32 + 48z) − 1 + 1 + y [32 + 48z + x(48 + 72z) − 1 + 1] = 22 + 32z + x(32 + 48z) + y [32 + 48z + x(48 + 72z)] = |E₃(T₇)|. We have completed the proof of this case. Figure 6 is an illustration for x = y = z = 1, that is, T_10,10,10.

The remaining cases, involving the construction of Hamiltonian cycles on B₉ through B₃₆, is summarized in Figure 7. □

4. Conclusions

We investigate the existence of WDBHs in three-dimensional toroidal mesh graphs T_m,n,r for all m, n, r ≥ 3. We merge two Hamiltonian cycles by removing one or two edges from each and reconnecting them. When removing one edge per cycle, the new Hamiltonian cycle is straightforward to construct; when removing two edges, careful connection is essential to avoid disconnected cycles (which need to satisfy Lemma 2). We process the ten cases where DBHs are known not to exist. By applying the construction method in Figure 4 and using the B₁ to B₃₆ cycles shown in Figure 5 and Figure 7, along with Table 2, we successfully built a WDBH for each case. This confirms that a WDBH exists in every three-dimensional toroidal mesh graph. This is an important result, especially compared with the two-dimensional case, where WDBHs are known to be absent in several configurations. It raises a compelling question for future research on a higher-dimensional toroidal mesh graph with WDBH. This direction offers a promising avenue for further theoretical exploration.