On $\sigma$-Tripartite Labelings of Odd Prisms and Even Möbius Ladders

Abstract

A common question in the study of graph decompositions is when does a graph G decompose the complete graph or the complete graph with a 1-factor removed or added. It is known that a $\sigma$-tripartite labeling of a tripartite graph G with n edges can be used to obtain a cyclic G-decomposition of $K_{2nt+1}$ for every positive integer t. Moreover, it can be used to obtain a cyclic G-decomposition of both $K_{2nt+2}-I$ and $K_{2nt}+I$, where I is a 1-factor. We show that if G is an odd prism on 10 or more vertices or an even Möbius ladder, then G admits a $\sigma$-tripartite labeling.

1 Introduction

For integers r and s, we denote the set $\{r, r+1, \dots , s\}$ by [r, s] (if $r>s$, then $[r,s]=\emptyset$). Let $\mathbb {N}$ denote the set of nonnegative integers, $\mathbb {Z}^{+}$ denote the set of positive integers, and $\mathbb {Z}_n$ denote the group of integers modulo n. Call a graph Gtripartite if the chromatic number of G is at most 3. Thus, bipartite graphs can be considered tripartite.

Let m be a positive integer and let $V(K_m)=[0, m-1]$. The length of an edge $\{i,j\}$ in $K_m$ is $\min \{|i-j|,m-|i-j|\}$. Note that if m is odd, then $K_m$ consists of m edges of length i for $i\in [1,\frac{m-1}{2}]$. However, if m is even, then $K_m$ consists of m edges of length i for $i\in [1,\frac{m}{2}-1]$ and of only $\frac{m}{2}$ edges of length $\frac{m}{2}$. In this case, the edges of length $\frac{m}{2}$ form a 1-factor in $K_m$. Throughout this manuscript, if m is even, we will denote the 1-factor formed by the set of edges of length $\frac{m}{2}$ in $K_m$ by I.

Let $V(K_m)=\mathbb {Z}_m$ and let G be a subgraph of $K_m$. By clickingG, we mean applying the permutation $i\mapsto i+1$ to V(G). Note that clicking an edge does not change its length. Let H and G be graphs such that G is a subgraph of H. A G-decomposition of H is a set $\Delta =\{G_1,G_2,\dots ,G_t\}$ of pairwise edge-disjoint subgraphs of H each of which is isomorphic to G and such that $E(H)=\bigcup _{i=1}^{t}E(G_i)$. A G-decomposition of $K_m$ is also known as a $(K_m,G)$-design. A $(K_m,G)$-design $\Delta$ is cyclic if clicking is an automorphism of $\Delta$. The study of graph decompositions is generally known as the study of graph designs, or G-designs. For surveys on G-designs, see [1] and [2].

Let G be a graph with n edges. A primary question in the study of graph designs is: for what values ofvdoes there exist a$(K_v,G)$-design? Another question of interest is the existence of $(K_v \pm I,G)$-designs where v is even. For most studied graphs G, it is often the case that if $v \equiv 1\pmod {2n}$, then there exists a $(K_v,G)$-design. Similarly, if $v \equiv 2 \pmod {2n}$ or $v \equiv 0 \pmod {2n}$, then there often exists a $(K_v-I,G)$-design in the former and a $(K_v+I,G)$-design in the latter. A common approach to finding these designs is through the use of graph labelings.

1.1 Graph Labelings

For a graph G, a one-to-one function $f:V(G)\rightarrow \mathbb {N}$ is called a labeling (or a valuation) of G. In a seminal paper on the topic [11], Rosa introduced a hierarchy of labelings. Let G be a graph with n edges and no isolated vertices and let f be a labeling of G. Let $f(V(G))=\{f(u): u \in V(G)\}$. Define a function $\bar{f}:E(G) \rightarrow \mathbb {Z}^+$ by $\bar{f}(e)=|f(u)-f(v)|$, where $e=\{u,v\} \in E(G)$. We will refer to $\bar{f}(e)$ as the label of e. Let $\bar{f}(E(G))=\{\bar{f}(e): e\in E(G)\}$. Consider the following conditions:

($\ell 1$):

$f(V(G)) \subseteq [0,2n]$,

($\ell 2$):

$f(V(G)) \subseteq [0,n]$,

($\ell 3$):

$\bar{f}(E(G)) = \{x_1, x_2, \ldots , x_n\}$, where for each $i \in [1,n]$ either $x_i=i$ or $x_i=2n+1-i$,

($\ell 4$):

$\bar{f}(E(G)) = [1,n]$.

If in addition G is bipartite with bipartition $\{A, B\}$ of V(G) consider also

($\ell 5$):

for each $\{a,b\} \in E(G)$ with $a \in A$ and $b \in B$, we have $f(a)<f(b)$,

($\ell 6$):

there exists an integer $\lambda$ such that $f(a) \le \lambda$ for all $a \in A$ and $f(b) > \lambda$ for all $b \in B$.

Then a labeling satisfying the conditions:

$(\ell 1), (\ell 3)$ :

is called a $\rho$-labeling;

$(\ell 1), (\ell 4)$ :

is called a $\sigma$-labeling;

$(\ell 2), (\ell 4)$ :

is called a $\beta$-labeling.

A $\beta$-labeling is necessarily a $\sigma$-labeling, which in turn is a $\rho$-labeling. Suppose G is bipartite. If a $\rho$-, $\sigma$-, or $\beta$-labeling of G satisfies condition $(\ell 5)$, then the labeling is ordered and is denoted by $\rho ^+$, $\sigma ^+$, or $\beta ^+$, respectively. If in addition $(\ell 6)$ is satisfied, the labeling is uniformly ordered and is denoted by $\rho ^{++}$, $\sigma ^{++}$, or $\beta ^{++}$, respectively.

A $\beta$-labeling is better known as a graceful labeling, and a uniformly ordered $\beta$-labeling is an $\alpha$-labeling as introduced in [11]. Labelings of the types above are called Rosa-type labelings because of Rosa’s original article [11] on the topic. (See [4] for a recent comprehensive survey of Rosa-type labelings.) A dynamic survey on general graph labelings is maintained by Gallian [8].

Labelings are critical to the study of cyclic graph decompositions as seen in the following theorem from [11].

Theorem 1

Let G be a graph with n edges. There exists a cyclic G-decomposition of $K_{2n+1}$ if and only if G admits a $\rho$-labeling.

If G admits a $\sigma$-labeling instead, then cyclic G-decompositions of $K_{2n+2}-I$ and of $K_{2n}+I$ can also obtained. Theorem 2 appears as Theorem 3.5 in [4]. We provide a proof of Theorem 3 for the sake of completeness.

Theorem 2

Let G be a graph with n edges. If G admits a $\sigma$-labeling, then there also exists a cyclic G-decomposition of $K_{2n+2}-I$.

Theorem 3

Let G be a graph with n edges. If G admits a $\sigma$-labeling, then there exists a cyclic G-decomposition of $K_{2n}+I$.

Proof

Let G, n and I be as in the statement of the theorem. Let $V(K_{2n})=\mathbb {Z}_{2n}$. Note that $K_{2n}+I$ is the multigraph obtained form $K_{2n}$ by making each of the edges of length n have multiplicity 2. Thus, for each $i \in [1,n]$, the number of edges of length i in $K_{2n}+I$ is 2n. Let h be a $\sigma$-labeling of G. Let $G_0$ be the subgraph of $K_{2n}+I$ obtained by identifying vertex $v \in V(G)$ with $i \in V(K_{2n})$ if $h(v)=i$. Thus, $G_0$ is an embedding of G in $K_{2n}$ so that there is an edge in $G_0$ of length i for each $i\in [1,n]$. For $t\in [1,2n-1]$, let $G_t$ be the subgraph of $K_{2n}+I$ obtained by clicking $G_0$ a total of t times. Then $\Delta =\{G_t: t \in \mathbb {Z}_{2n}\}$ is a cyclic G-decomposition of $K_{2n}+I$. $\square$

If G admits an $\alpha$-labeling, then we have the following powerful result of Rosa [11].

Theorem 4

Let G be a bipartite graph with n edges. If G admits an $\alpha$-labeling, then there exists a cyclic G-decomposition of $K_{2nt+1}$ for all positive integers t.

We illustrate how Theorem 4 works. Let h be an $\alpha$-labeling of a graph G with n edges and bipartition (A, B). Let $A=\{u_1, u_2, \ldots , u_r\}$ and $B=\{v_1, v_2, \ldots , v_s\}$. Let t be a positive integer. For $1 \le i \le t$, let $G_i$ be a copy of G with bipartition $(A,B_i)$ where $B_i=\{v_{i,1}, v_{i,2}, \ldots , v_{i,s}\}$ and $v_{i,j}$ corresponds to $v_j$ in B. Let $G(t)=G_1 \cup G_2 \cup \ldots \cup G_t$. Thus, G(t) has nt edges and is bipartite with bipartition $(A,B_1 \cup B_2 \cup \ldots \cup B_t)$. Define a labeling $f'$ of G(t) as follows: $f'(u_j)=f(u_j)$ for each $u_j \in A$ and $f'(v_{i,j})=f(j)+(i-1) n$ for $1 \le i \le t$ and $1 \le j \le s$. It is easy to see that $f'$ is an $\alpha$-labeling of G(t), and thus, Theorem 1 applies. Since $f'$ is necessarily a $\sigma$-labeling, Theorems 2 and 3 also apply, and we have the following.

Corollary 5

Let G be a bipartite graph with n edges. If G admits an $\alpha$-labeling, then there exist cyclic G-decompositions of $K_{2nt+2}-I$ and of $K_{2nt}+I$ for all positive integers t.

From a graph decompositions perspective, Theorem 2 offers a slight advantage over Theorem 1. In the case G is bipartite, Theorem 4 offers a great advantage over the first two. However, there are many classes of bipartite graphs (see [4]) that do not admit $\alpha$-labelings. Theorem 4 was extended to cover graphs that admit $\rho ^+$-labelings in [5].

Theorem 6

Let G be a bipartite graph with n edges. If G admits a $\rho ^+$-labeling, then there exists a cyclic G-decomposition of $K_{2nt+1}$ for all positive integers t.

Again, since Theorem 6 is $\rho$-labeling based, it does not guarantee the existence of decomposition results involving the addition or removal of a 1-factor. Two labelings that lead to results similar to those of Theorems 4 and 6 were recently introduced for tripartite graphs in [3]. One of them is called a $\rho$-tripartite labeling and the other a $\sigma$-tripartite labeling. Both lead to cyclic G-decompositions of $K_{2nt+1}$, but only the $\sigma$-labeling-based one leads to results involving the addition or removal of a 1-factor. In this manuscript, we focus on the $\sigma$-labeling-based one.

Let G be a tripartite graph with n edges having the vertex tripartition $\{A, B, C\}$. A $\sigma$-tripartite labeling of G is a one-to-one function $h:V(G) \rightarrow [0, 2n]$ that satisfies the following conditions:

(s1):

h is a $\sigma$-labeling of G.

(s2):

If $\{a,v\} \in E(G)$ with $a \in A$, then $h(a)<h(v)$.

(s3):

If $e=\{b,c\} \in E(G)$ with $b \in B$ and $c \in C$, then there exists an edge $e' =\{b',c'\} \in E(G)$ with $b' \in B$ and $c' \in C$ such that $|h(c')-h(b')|+|h(c)-h(b)| = n$.

(s4):

If $a \in A$ and $v \in B \cup C$, then $h(a)-h(v) \ne n$.

(s5):

If $b \in B$ and $c \in C$, then $|h(b)-h(c)| \not \in \{n, 2n\}$.

Note that e and $e'$ in (s3) need not to be distinct. Also note that there need not be an edge $\{a,v\}$ in (s4) nor an edge $\{b,c\}$ in (s5). The following theorem from [3] shows that a $\sigma$-tripartite labeling yields results similar to those from $\alpha$-labelings.

Theorem 7

Let G be a tripartite graph with n edges. If G admits a $\sigma$-tripartite labeling, then there exists a cyclic G-decomposition of $K_{2nt+1}$ for all positive integers t.

Again, we illustrate how Theorem 7 works. Let G have n edges and let h be a $\sigma$-tripartite labeling of G with vertex tripartition $\{A, B, C\}$ as in the above definition. Let $B_{1}, B_{2}, \ldots , B_{t}$ be t vertex-disjoint copies of B, and let $C_{1}, C_{2}, \ldots ,C_{t}$ be t vertex-disjoint copies of C. The vertex in $B_{i}$ corresponding to $b \in B$ will be called $b_{i}$. Similarly, the vertex in $C_{i}$ corresponding to $c \in C$ will be called $c_{i}$. Let $B^{*}=\bigcup _{i=1}^{t}B_{i}$ and $C^{*}=\bigcup _{i=1}^{t}C_{i}$. We define a new graph $G^{*}$ with vertex set $A \bigcup B^{*} \bigcup C^{*}$ and edges $\{a,v_{i}\}$, $1 \le i \le t$, whenever $a\in A$ and $\{a,v\}$ is an edge of G, and $\{{b_{i}},c_{i}\}$, $1 \le i \le t$, whenever $\{b,c\}$ is an edge of G with $b \in B$ and $c\in C$. Clearly $G^{*}$ has nt edges and G divides $G^{*}$. Define a labeling $h^{*}$ on $G^{*}$ by

$$\begin{aligned} h^{*}(v) = {\left\{ \begin{array}{ll} h(v) &{} v\in A,\\ h(b)+(i-1) n &{} v=b_{i}\in B_{i},\\ h(c)+(t-i) n &{} v=c_{i} \in C_{i}. \end{array}\right. } \end{aligned}$$

The labeling $h^*$ is a $\sigma$-labeling of $G^*$ and the result follows by Theorem 1. Moreover, we can use Theorems 2 and 3 to obtain cyclic G-decompositions of $K_{2nt+2}-I$ and of $K_{2nt}+I$ as well. The $K_{2nt+2}-I$ result appears as Corollary 5 in [3].

Corollary 8

Let G be a graph with n edges. If G admits a $\sigma$-tripartite labeling, then there exist cyclic G-decompositions of $K_{2nt+2}-I$ and of $K_{2nt}+I$ for every positive integer t.

In Fig. 1, we demonstrate the use of Theorem 7 in the case $t=3$ by showing a $\sigma$-tripartite labeling of the graph G consisting of a triangle with a pendent edge. The labeling of G on the left can be used to yield cyclic G-decompositions of $K_{9}$, of $K_{8}+I$, and of $K_{10}-I$. The labeling of the three copies of G on the right can be used to yield cyclic G-decompositions of $K_{25}$, of $K_{24}+I$, and of $K_{26}-I$.

Fig. 1

A $\sigma$-tripartite labeling of a graph G with 4 edges and the 3 copies of G used to yield cyclic G-decompositions of $K_{25}$, of $K_{24}+I$, and of $K_{26}-I$

Some Rosa-type labelings of various cubic graphs have been investigated. It is known that all bipartite prisms [6, 7] and bipartite Möbius ladders [9] admit $\alpha$-labelings. In [17], it is shown that if G is cubic and bipartite and if every component of G is either a prism, a Möbius ladder, or has order at most 14, then G admits an $\alpha$-labeling. Hence, if such a bipartite G has n edges, then it cyclically decomposes $K_{2nt+1}$, $K_{2nt}+I$, and $K_{2nt+2}-I$ for every positive integer t. In [16], it is shown that if G is an odd prism, an even Möbius ladder, or a connected cubic tripartite graph of order at most 10, then G admits a $\rho$-tripartite labeling. Hence, such a G of size n would cyclically decompose $K_{2nt+1}$ for every positive integer t. However, no G-decompositions of $K_{2nt}+I$ or $K_{2nt+2}-I$ can be obtained from this labeling. In [15], it is shown that every cubic graph of order at most 12, other than $2K_4$ and $3K_4$, admits a $\beta$-labeling. Vietri [13, 14] has shown that certain classes of generalized Petersen graphs are graceful. It is also known that $2K_4$ does not admit a $\rho$-labeling, but $3K_4$ does.

In this article, we show that if G is an odd prism on 10 or more vertices or an even Möbius ladder, then G admits a $\sigma$-tripartite labeling, and hence, such a G of size n would cyclically decompose $K_{2nt}+I$ and $K_{2nt+2}-I$, in addition to $K_{2nt+1}$, for every positive integer t.

1.2 Additional Definitions and Notation

We denote the path with vertices $x_0, x_1, \ldots , x_k$, where $x_i$ is adjacent to $x_{i+1}$, $0 \le i \le k-1$, by $(x_0, x_1, \ldots , x_k)$. In using this notation, we are thinking of traversing the path from $x_0$ to $x_k$ so that $x_0$ is the first vertex, $x_1$ is the second vertex, and so on. Let $G_1 =(x_0, x_1, \ldots , x_j)$ and $G_2=(y_0, y_1, \ldots , y_k)$. If $G_1$ and $G_2$ are vertex-disjoint except for $x_j=y_0$, then by $G_1+G_2$ we mean the path $(x_0, x_1, \ldots , x_j, y_1, y_2, \ldots , y_k)$. If the only vertices they have in common are $x_0=y_k$ and $x_j=y_0$, then by $G_1+G_2$ we mean the cycle $(x_0, x_1, \ldots , x_j, y_1, y_2, \ldots , y_{k-1}, x_0)$.

Let P(2k) be the path with 2k edges and $2k+1$ vertices $0, 1, \ldots , 2k$ given by $(0, 2k, 1, 2k-1, 2, 2k-2, \ldots , k-1,k+1,k)$. Note that the set of vertices of this graph is $A\cup B$, where $A = [0,k ]$, $B = [k+1,2k]$, and every edge joins a vertex from A to one from B. Furthermore, the set of labels of the edges of P(2k) is [1, 2k].

Let a and b be nonnegative integers and k, $d_1$, and $d_2$ be positive integers such that $a+kd_1<b$. Let $\hat{P}(2k, d_1,d_2,a,b)$ be the path with 2k edges and $2k+1$ vertices given by $(a, b+(k-1)d_2, a+d_1, b+(k-2)d_2, a+2d_1, \ldots , a+(k-1)d_1,b,a+kd_1)$. Note that $\hat{P}(2k,1,1,0,k+1)$ is the graph P(2k). Note that this graph $\hat{P}(2k, d_1,d_2,a,b)$ has the following properties:

P1::

$\hat{P}(2k, d_1,d_2,a,b)$ is a path with first vertex a, second vertex $b+(k-1)d_2$, and last vertex $a+kd_1$.

P2::

Each edge of $\hat{P}(2k, d_1,d_2,a,b)$ joins a vertex from $A = \{a+id_1:0 \le i \le k\}$ to a vertex with a larger label from $B =\{b+id_2: 0 \le i \le k-1\}$.

P3::

The set of edge labels of $\hat{P}(2k, d_1,d_2,a,b)$ is $\{b-a-kd_1+i(d_1+d_2): 0 \le i \le k-1\} \cup \{b-a-(k-1)d_1+i(d_1+d_2): 0 \le i \le k-1\}$.

The path $\hat{P}(12,2,4,10,30)$ is shown in Fig. 2 below.

Fig. 2

Path $\hat{P}(12,2,4,10,30)$

2 $\sigma$-Tripartite Labelings of Some Cubic Graphs

We will show that odd prisms and even Möbius ladders admit $\sigma$-tripartite labelings.

2.1 $\sigma$-Tripartite Labelings of Odd Prisms

By the prism$D_{n}\ (n\ge 3)$ we mean the Cartesian product of a cycle with n vertices and a path with 2 vertices: $C_{n}\times P_{2}$. For convenience, we let $D_n=C_n \cup C'_n \cup F$, where $C_n=(v_{1},v_{2},\ldots ,v_{n},v_1)$, $C'_n=(v'_{1},v'_{2},\ldots ,v'_{n},v'_1)$, and $F=\{\{v_i,v'_i\}: 1 \le i\le n\}$. We shall refer to $C_n$ as the outer cycle, to $C'_n$ as the inner cycle, and to F as the spokes. We note that $D_{2n+1}$ (for $n > 1$) is necessarily tripartite with tripartition $\{A, B, C\}$ where $A= \{v'_1\} \cup \{v_{2i+1}: 2\le i \le n\} \cup \{v'_{2i}: 2\le i \le n\}$, $B= \{v_{2i}: 1 \le i \le n\} \cup \{ v'_{2i+1}: 1\le i \le n\}$, and $C= \{v_1, v_3, v'_2\}$. Figure 3 shows the prism $D_7$. In this figure, the vertices in A are shown with white circles while the vertices in B are shown with black circles and the vertices of C are shown with white squares. The edges between sets B and C are shown in thick lines. We will adopt this convention in all our figures. It is easy to see that $D_3$ cannot admit a $\sigma$-tripartite labeling. We will show that $D_{n}$ admits a $\sigma$-tripartite labeling for all odd integers $n \ge 5$.

Fig. 3

Prism $D_7$

Lemma 9

The prism $D_3$ does not admits a $\sigma$-tripartite labeling.

Proof

If $\{A,B,C\}$ is a vertex tripartition of $D_3$, then the number of edges between B and C is necessarily 3. Since the number of edges of $D_3$ is odd, it is impossible for $D_3$ to admit a $\sigma$-tripartite labeling.

Lemma 10

The prism $D_n$ admits a $\sigma$-tripartite labeling for all $n \in \{5, 7, 9\}$.

Proof

We give $\sigma$-tripartite labelings of $D_5$, $D_7$, and $D_9$ in Fig. 4. $\square$

Fig. 4

A $\sigma$-tripartite labeling of $D_{5}, D_7$ and $D_9$

Theorem 11

The prism $D_n$ admits a $\sigma$-tripartite labeling for all odd $n\ge 5$.

Proof

The cases with $n\le 9$ are covered in Lemma 10. We separate the rest of the proof into 3 cases.

Case 1$n \equiv 1 \pmod {6}$.

Let $n=6t+1$ where $t\ge 2$. Thus, $|V(D_{n})|=12t+2$ and $|E(D_{n})|=18t+3$. Define a one-to-one function $f:V(D_{6t+1}) \rightarrow [0,36t+6]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t-1,&\\ f(v_{2})&= 18t,&\\ f(v_{3})&= 36t+2,&\\ f(v_{4})&= 18t+2,&\\ f(v_{i})&= i+2,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t-1\}, \\ f(v_{i})&= 18t-2i+10,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 6 \le i \le 2t+4\},\\ f(v_{i})&= 18t-2i+4,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t+4< i \le 6t-2\},\\ f(v_{6t})&= 18t-5,&\\ f(v_{6t+1})&= 1,&\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t+3,&\\ f(v'_{3})&= 18t+1,&\\ f(v'_{4})&= 5,&\\ f(v'_{5})&= 12t-3,&\\ f(v'_{i})&= i+2,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 6 \le i \le 6t\},\\ f(v'_{i})&= 18t-2i+10,&v'_i \in B'_1=\{v'_i: i\ \mathrm{odd},\ 7 \le i \le 2t+3\},\\ f(v'_{i})&= 18t-2i+4,&v'_i \in B'_2=\{v'_i: i\ \mathrm{odd},\ 2t+3 < i \le 6t-1\},\\ f(v'_{6t+1})&= 18t-7.&\end{aligned}$$

Note that $A=\{v_{6t+1}, v'_1, v'_4\}\cup A_1\cup A'_1$, $B=\{v_2, v_4, v_{6t}, v'_3, v'_5, v'_{6t+1} \} \cup B_1 \cup B_2 \cup B'_1 \cup B'_2$ and $C=\{v_1, v_3, v'_2\}$. Thus, the domain of f is indeed $V(D_{6t+1})$. Next, we confirm that f is one-to-one. We compute

$$\begin{aligned}&f(A_1) = \{7, 9, \ldots , 6t+1\},\\&f(A'_1) = \{8, 10, \ldots , 6t+2\},\\&f(B_1) = \{18t-2, 18t-6, \ldots , 14t+2\},\\&f(B_2) = \{14t-8, 14t-12, \ldots , 6t+8\},\\&f(B'_1) = \{18t-4, 18t-8, \ldots , 14t+4\},\\&f(B'_2) = \{14t-6, 14t-10, \ldots , 6t+6\}. \end{aligned}$$

Note that f is piecewise strictly increasing by 2 or strictly decreasing by 4 and that all labels are distinct. Thus, f is one-to-one. Moreover, $f(A)\subseteq [0,6t+2]$ and $f(B \cup C) \subseteq [6t+6, 36t+2]$.

To help compute the edge labels, we will describe $f(V(D_{6t+1}))$ in terms of the $\hat{P}(2k,d_1,d_2,a,b)$ path notation. For convenience, we will identify the vertices of $C_{6t+1}$ and $C'_{6t+1}$ with their labels. We have $f(C_{6t+1})=G_1+G_2+(6t+1, 18t-5, 1, 18t-1, 18t, 36t+2, 18t+2, 7)$, where

$$\begin{aligned}&G_1=\hat{P}(2(t),2,4,7,14t+2),\\&G_2=\hat{P}(2(2t-3),2,4,2t+7,6t+8). \end{aligned}$$

By P3, the resulting edge label sets are:

$$\begin{aligned} \bar{f}(E(G_1))&= \{12t-5+6i: 0 \le i \le t-1\} \cup \{12t-3+6i: 0 \le i \le t-1\}\\&= \{\ell \equiv {1}\ \mathrm{(mod\ 6)}: 12t-5 \le \ell \le 18t-11\} \\ {}&\qquad \cup \{\ell \equiv {3}\ \mathrm{(mod\ 6)}: 12t-3 \le \ell \le 18t-9\},\\ \bar{f}(E(G_2))&= \{7+6i: 0 \le i \le 2t-4\} \cup \{9+6i: 0 \le i \le 2t-4\}\\&= \{\ell \equiv {1}\ \mathrm{(mod\ 6)}: 7 \le \ell \le 12t-17\} \\ {}&\qquad \cup \{\ell \equiv {3}\ \mathrm{(mod\ 6)}: 9 \le \ell \le 12t-15\}. \end{aligned}$$

Moreover, edge labels $12t-6, 18t-6, 18t-2, 1, 18t+2, 18t,$ and $18t-5$ occur on the path $(6t+1, 18t-5, 1, 18t-1, 18t, 36t+2, 18t+2, 7)$.

Similarly, we have $f(C'_{6t+1})=G'_1+G'_2+(6t+2, 18t-7, 0, 18t+3, 18t+1, 5,12t-3,8)$, where

$$\begin{aligned}&G'_1=\hat{P}(2(t-1),2,4,8,14t+4),\\&G'_2=\hat{P}(2(2t-2),2,4,2t+6,6t+6). \end{aligned}$$

By P3, the resulting edge label sets are:

$$\begin{aligned} \bar{f}(E(G'_1))&= \{12t-2+6i: 0 \le i \le t-2\} \cup \{12t+6i: 0 \le i \le t-2\}\\&= \{\ell \equiv {4}\ \mathrm{(mod\ 6)}: 12t-2 \le \ell \le 18t-14\} \\ {}&\qquad \cup \{\ell \equiv {0}\ \mathrm{(mod\ 6)}: 12t \le \ell \le 18t-12\},\\ \bar{f}(E(G'_2))&= \{4+6i: 0 \le i \le 2t-3\} \cup \{6+6i: 0 \le i \le 2t-3\}\\&= \{\ell \equiv {4}\ \mathrm{(mod\ 6)}: 4 \le \ell \le 12t-14\} \\ {}&\qquad \cup \{\ell \equiv {0}\ \mathrm{(mod\ 6)}: 6 \le \ell \le 12t-12\}. \end{aligned}$$

Moreover, edge labels $12t-9$, $18t-7$, $18t+3$, 2, $18t-4$, $12t-8$, and $12t-11$ occur on the path $(6t+2, 18t-7, 0, 18t+3, 18t+1, 5, 12t-3, 8)$.

For each spoke $\{v_i,v'_i\}$, the labels on the spokes are given by

$$\begin{aligned} \bar{f}(\{v_i,v'_i\})={\left\{ \begin{array}{ll} 18t-1 &{} \mathrm{for}\ i=1,\\ 3 &{} \mathrm{for}\ i=2,\\ 18t+1 &{} \mathrm{for}\ i=3,\\ 18t-3 &{} \mathrm{for}\ i=4,\\ 12t-10 &{} \mathrm{for}\ i=5,\\ 18t-3i+8 &{} \mathrm{for}\ 6 \le i \le 2t+4,\\ 18t-3i+2 &{} \mathrm{for}\ 2t+5 \le i \le 6t-1,\\ 12t-7 &{} \mathrm{for}\ i=6t,\\ 18t-8 &{} \mathrm{for}\ i=6t+1. \end{array}\right. } \end{aligned}$$

Thus, the set of edge labels on the spokes is

$$\begin{aligned} \bar{f}(E(F))&= \{\ell \equiv {2}\ \mathrm{(mod\ 3)}: 12t-4 \le \ell \le 18t-10\} \\&\qquad \cup \{\ell \equiv {2}\ \mathrm{(mod\ 3)}: 5 \le \ell \le 12t-13\}\\&\qquad \cup \{ 18t-1, 3, 18t+1, 18t-3, 12t-10,12t-7,18t-8\}. \end{aligned}$$

It is easy to verify now that each $\ell \in [1,18t+3]$ occurs on exactly one edge in $D_{6t+1}$. Hence, the defined labeling is a $\sigma$-labeling, and condition (s1) for a $\sigma$-tripartite labeling is satisfied. Condition (s2) also holds since $f(A) \subseteq [0, 6t+2]$ and $f(B \cup C) \subseteq [6t+6, 36t+2]$. Condition (s3) holds since $|f(v_1)-f(v_2)| + |f(v_2)-f(v_3)|=18t+3$, $|f(v_3)-f(v_4)|+|f(v_2)-f(v'_2)|=18t+3$, and $|f(v_3)-f(v'_3)| + |f(v'_2)-f(v'_3)|=18t+3$, number of edges of $D_{6t+1}$. Condition (s4) clearly holds. Also $|f(b)-f(c)| \in \{18t+3, 36t+6\}$, where $b \in B$ and $c \in C$, is impossible since $|f(b)-f(c)| \in \{1, 2, 3, 18t, 18t+1, 18t+2\}$. Thus, condition (s5) holds, and we have a $\sigma$-tripartite labeling of $D_{6t+1}$. Figure 5 shows a $\sigma$-tripartite labeling of $D_{13}$.

Fig. 5

A $\sigma$-tripartite labeling of $D_{13}$

Case 2$n \equiv 3 \pmod {6}$.

Let $n=6t-3$ where $t\ge 3$. Thus, $|V(D_{n})|=12t-6$ and $|E(D_{n})|=18t-9$. Define a one-to-one function $f:V(D_{6t-3}) \rightarrow [0,36t-18]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t-9,&\\ f(v_{2})&= 18t-10,&\\ f(v_{3})&= 36t-22,&\\ f(v_{4})&= 18t-12,&\\ f(v_{i})&= i-1,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t-3\}, \\ f(v_{i})&= 18t-2i-4,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 6 \le i \le 2t\},\\ f(v_{i})&= 18t-2i-5,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t< i \le 4t-2\},\\ f(v_{i})&= 18t-2i-6,&v_i \in B_3=\{v_i:i\ \mathrm{even},\ 4t-2< i \le 6t-4\},\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t-13,&\\ f(v'_{3})&= 18t-11,&\\ f(v'_{i})&= i-1,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 4 \le i \le 6t-4\},\\ f(v'_{i})&= 18t-2i-4,&v'_i \in B'_1=\{v'_i: i\ \mathrm{odd},\ 5 \le i \le 2t-1\},\\ f(v'_{i})&= 18t-2i-5,&v'_i \in B'_2=\{v'_i: i\ \mathrm{odd},\ 2t-1< i \le 4t-3\},\\ f(v'_{i})&= 18t-2i-6,&v'_i \in B'_3=\{v'_i: i\ \mathrm{odd},\ 4t-3 < i \le 6t-3\}. \end{aligned}$$

Note that $A=\{v'_1\}\cup A_1\cup A'_1$, $B=\{v_2, v_4, v'_3 \} \cup B_1 \cup B_2 \cup B_3 \cup B'_1 \cup B'_2\cup B'_3$ and $C=\{v_1,v_3, v'_2\}$. If we proceed as in Case 1, it is easy to verify that we have a $\sigma$-tripartite labeling of $D_{6t-3}$. Figure 6 shows a $\sigma$-tripartite labeling of $D_{15}$.

Fig. 6

A $\sigma$-tripartite labeling of $D_{15}$

Case 3$n \equiv 5 \pmod {6}$.

Let $n=6t-1$ where $t\ge 2$. Thus, $|V(D_{n})|=12t-2$ and $|E(D_{n})|=18t-3$. Define a one-to-one function $f: V(D_{6t-1}) \rightarrow [0,36t-6]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t-7,&\\ f(v_{2})&= 18t-6,&\\ f(v_{3})&= 36t-10,&\\ f(v_{4})&= 18t-4,&\\ f(v_{i})&= i+2,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t-3\}, \\ f(v_{i})&= 18t-2i+1,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 6 \le i \le 2t+2\},\\ f(v_{i})&= 18t-2i-2,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t+2< i \le 6t-4\},\\ f(v_{6t-2})&= 18t-14,&\\ f(v_{6t-1})&= 1,&\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t-3,&\\ f(v'_{3})&= 18t-5,&\\ f(v'_{4})&= 5,&\\ f(v'_{i})&= i+2,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 6 \le i \le 6t-2\},\\ f(v'_{i})&= 18t-2i+1,&v'_i \in B'_1=\{v'_i: i\ \mathrm{odd}, 5 \le i \le 2t+3\},\\ f(v'_{i})&= 18t-2i-2,&v'_i \in B'_2=\{v'_i: i\ \mathrm{even}, 2t+3 < i \le 6t-3\},\\ f(v'_{6t-1})&= 18t-12. \end{aligned}$$

We have that $A=\{v_{6t-1}, v'_1, v'_4\}\cup A_1\cup A'_1$, $B=\{v_2, v_4,v_{6t-2}, v'_3,v'_{6t-1}\} \cup B_1 \cup B_2 \cup B'_1 \cup B'_2$ and $C=\{v_1, v_3, v'_2\}$. If we proceed as in case 1, it is easy to verify that we have a $\sigma$-tripartite labeling of $D_{6t-1}$. Figure 7 shows a $\sigma$-tripartite labeling of $D_{11}$. $\square$

Fig. 7

A $\sigma$-tripartite labeling of $D_{11}$

2.2 $\sigma$-Tripartite Labelings of Even Möbius Ladders

For $n\ge 3$, let $v_{1},v_{2},\ldots ,v_{n}$ and $v'_{1},v'_{2},\ldots ,v'_{n}$ denote the consecutive vertices of two disjoint paths with n vertices. The Möbius ladder$M_n$ is the graph obtained by joining $v_{i}$ to $v'_{i}$ for $i=1,2,\ldots ,n$ and by joining $v_{1}$ to $v'_{n}$ and $v_{n}$ to $v'_{1}$. For convenience, we let $M_n=P_n \cup P'_n \cup F \cup H$, where $P_n=(v_{1},v_{2},\ldots ,v_{n})$, $P'_n=(v'_{1},v'_{2},\ldots ,v'_{n})$, $F=\{\{v_i,v'_i\}: 1 \le i\le n\}$ and $H=\{\{v_1,v'_n \}, \{v_n,v'_1\}\}$. We shall refer to $P_n$ as the outer path, to $P'_n$ as the inner path, and to F as the spokes. Figure 8 shows the Möbius ladder $M_{10}$. We note that $M_{2n}$ (with $n \ge 2$) is necessarily tripartite with tripartition $\{A, B, C\}$, where $A =\{v'_1, v'_4\} \cup \{v_{2i-1}, v'_{2i} : 3\le i \le n\}$, $B = \{v_2\} \cup \{ v'_{2i-1}, v_{2i} : 2 \le i \le n\}$, and $C = \{ v_1, v_3, v'_2\}$. We will show that $M_{n}$ admits a $\sigma$-tripartite labeling for all even integers $n \ge 4$.

Fig. 8

Möbius ladder $M_{10}$

Lemma 12

The Möbius ladder $M_n$ admits a $\sigma$-tripartite labeling for all $n \in \{4, 6, 8, 10,12\}$.

Proof

We give $\sigma$-tripartite labelings of $M_4$, $M_6$, $M_8$, $M_{10}$, and $M_{12}$ in Fig. 9. $\square$

Fig. 9

$\sigma$-tripartite labelings of $M_4$, $M_6$, $M_8$, $M_{10}$, and $M_{12}$

Theorem 13

The Möbius ladder $M_n$ admits a $\sigma$-tripartite labeling for all even $n \ge 4$.

Proof

The cases with $n \le 12$ are covered in Lemma 12. We separate the rest of the proof into 3 cases.

Case 1$n \equiv 0 \pmod {6}$.

Let $n=6t$ where $t \ge 3$. Thus, $|V(M_{n})|=12t$ and $|E(M_{n})|=18t$. Define a one-to-one function $f :V(M_{6t}) \rightarrow [0,36t]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t,&\\ f(v_{2})&= 18t-1,&\\ f(v_{3})&= 36t-4,&\\ f(v_{i})&= i-1,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t-1\}, \\ f(v_{i})&= 18t-2i+5,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 4 \le i \le 2t\},\\ f(v_{i})&= 18t-2i+4,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t< i \le 4t\},\\ f(v_{i})&= 18t-2i+3,&v_i \in B_3=\{v_i:i\ \mathrm{even},\ 4t< i \le 6t\},\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t-4,&\\ f(v'_{3})&= 18t-2,&\\ f(v'_{i})&= i-1,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 4 \le i \le 6t\},\\ f(v'_{i})&= 18t-2i+5,&v'_i \in B'_1=\{v'_i:i\ \mathrm{odd},\ 5 \le i \le 2t+1\},\\ f(v'_{i})&= 18t-2i+4,&v'_i \in B'_2=\{v'_i:i\ \mathrm{odd},\ 2t+1< i \le 4t-1\},\\ f(v'_{i})&= 18t-2i+3,&v'_i \in B'_3=\{v'_i:i\ \mathrm{odd},\ 4t-1 < i \le 6t-1\}. \end{aligned}$$

Note that $A = \{v'_1\} \cup A_1 \cup A'_1$, $B =\{v_2, v'_3 \} \cup B_1 \cup B_2 \cup B_3 \cup B'_1 \cup B'_2 \cup B'_3$, and $C = \{v_1, v_3, v'_2\}$. Thus, the domain of f is indeed $V(M_{6t})$. Next, we confirm that f is one-to-one. We compute

$$\begin{aligned} f(A_1)&= \{4, 6, \ldots , 6t-2\},\\ f(A'_1)&= \{3, 5, \ldots , 6t-1\},\\ f(B_1)&= \{18t-3, 18t-7, \ldots , 14t+5\},\\ f(B_2)&= \{14t, 14t-4, \ldots , 10t+4\},\\ f(B_3)&= \{10t-1, 10t-5, \ldots , 6t+3\},\\ f(B'_1)&= \{18t-5, 18t-9, \ldots , 14t+3\},\\ f(B'_2)&= \{14t-2, 14t-6, \ldots , 10t+6\},\\ f(B'_3)&= \{10t+1, 10t-3, \ldots , 6t+5\}. \end{aligned}$$

Note that f is piecewise strictly increasing by 2 or strictly decreasing by 4 and that all labels are distinct. Thus, f is one-to-one. Moreover, $f(A)\subseteq [0,6t-1]$ and $f(B \cup C) \subseteq [6t+3, 36t-4]$.

To help compute the edge labels, we will describe $f(M_{6t})$ in terms of the $\hat{P}(2k,d_1,d_2,a,b)$ path notation. For convenience, we will identify the vertices of $P_{6t}$ and $P'_{6t}$ with their labels. We have $f(P_{6t})=(18t, 18t-1, 36t-4, 18t-3, 4)+G_1+G_2+G_3+(6t-2, 6t+3)$, where

$$\begin{aligned}&G_1=\hat{P}(2(t-2),2,4,4,14t+5),\\&G_2=\hat{P}(2(t),2,4,2t,10t+4),\\&G_3=\hat{P}(2(t-1),2,4,4t,6t+7). \end{aligned}$$

By P3, the resulting edge label sets are:

$$\begin{aligned} \bar{f}(E(G_1))&= \{12t+5+6i: 0 \le i \le t-3\} \cup \{12t+7+6i: 0 \le i \le t-3\}\\&= \{\ell \equiv {5}\ \mathrm{(mod\ 6)}: 12t+5 \le \ell \le 18t-13\} \\ {}&\qquad \cup \{\ell \equiv {1}\ \mathrm{(mod\ 6)}: 12t+7 \le \ell \le 18t-11\},\\ \bar{f}(E(G_2))&= \{6t+4+6i: 0 \le i \le t-1\} \cup \{6t+6+6i: 0 \le i \le t-1\}\\&= \{\ell \equiv {4}\ \mathrm{(mod\ 6)}: 6t+4 \le \ell \le 12t-2\} \\ {}&\qquad \cup \{\ell \equiv {0}\ \mathrm{(mod\ 6)}: 6t+6 \le \ell \le 12t\},\\ \bar{f}(E(G_3))&= \{9+6i: 0 \le i \le t-2\} \cup \{11+6i: 0 \le i \le t-2\}\\&= \{\ell \equiv {3}\ \mathrm{(mod\ 6)}: 9 \le \ell \le 6t-3\} \\ {}&\qquad \cup \{\ell \equiv {5}\ \mathrm{(mod\ 6)}: 11 \le \ell \le 6t-1\}. \end{aligned}$$

Moreover, edge labels 1, $18t-3$, $18t-1$, and $18t-7$ occur on the path $(18t, 18-1, 36t-4, 18t-3, 4)$ and the edge label 5 occurs on the edge $\{6t-2, 6t+3\}$.

Similarly, we have $f(P'_{6t})=(0, 18t-4, 18t-2, 3)+G'_1+G'_2+ G'_3$, where

$$\begin{aligned}&G'_1=\hat{P}(2(t-1),2,4,3,14t+3),\\&G'_2=\hat{P}(2(t-1),2,4,2t+1,10t+6),\\&G'_3=\hat{P}(2(t),2,4,4t-1,6t+5). \end{aligned}$$

By P3, the resulting edge label sets are:

$$\begin{aligned} \bar{f}(E(G'_1))&= \{12t+2+6i: 0 \le i \le t-2\} \cup \{12t+4+6i: 0 \le i \le t-2\}\\&= \{\ell \equiv {2}\ \mathrm{(mod\ 6)}: 12t+2 \le \ell \le 18t-10\} \\ {}&\qquad \cup \{\ell \equiv {4}\ \mathrm{(mod\ 6)}: 12t+4 \le \ell \le 18t-8\},\\ \bar{f}(E(G'_2))&= \{6t+7+6i: 0 \le i \le t-2\} \cup \{6t+9+6i: 0 \le i \le t-2\}\\&= \{\ell \equiv {1}\ \mathrm{(mod\ 6)}: 6t+7 \le \ell \le 12t-5\} \\ {}&\qquad \cup \{\ell \equiv {3}\ \mathrm{(mod\ 6)}: 6t+9 \le \ell \le 12t-3\},\\ \bar{f}(E(G'_3))&= \{6+6i: 0 \le i \le t-1\} \cup \{8+6i: 0 \le i \le t-1\}\\&= \{\ell \equiv {0}\ \mathrm{(mod\ 6)}: 6 \le \ell \le 6t\} \\ {}&\qquad \cup \{\ell \equiv {2}\ \mathrm{(mod\ 6)}: 8 \le \ell \le 6t+2\}. \end{aligned}$$

Moreover, edge labels $18t-4$, 2, and $18t-5$ occur on the path $(0, 18t-4, 18t-2, 3)$.

For each spoke $\{v_i,v'_i\}$, the labels on the spokes are given by

$$\begin{aligned} \bar{f}(\{v_i,v'_i\})={\left\{ \begin{array}{ll} 18t &{} \mathrm{for}\ i=1,\\ 3 &{} \mathrm{for}\ i=2,\\ 18t-2 &{} \mathrm{for}\ i=3,\\ 18t-3i+6 &{} \mathrm{for}\ 4 \le i \le 2t+1,\\ 18t-3i+5 &{} \mathrm{for}\ 2t+1< i \le 4t,\\ 18t-3i+4 &{} \mathrm{for}\ 4t < i \le 6t. \end{array}\right. } \end{aligned}$$

Thus, the set of edge labels on the spokes is

$$\begin{aligned} \bar{f}(E(F))&= \{\ell \equiv {0}\ \mathrm{(mod\ 3)}: 12t+3 \le \ell \le 18t-6\} \\ {}&\qquad \cup \{\ell \equiv {2}\ \mathrm{(mod\ 3)}: 6t+5 \le \ell \le 12t-1\} \\&\qquad \cup \{\ell \equiv {1}\ \mathrm{(mod\ 3)}: 4 \le \ell \le 6t+1\} \cup \{18t, 3, 18t-2\}. \end{aligned}$$

Moreover, edge labels $12t+1$ and $6t+3$ occur on the edges $\{v_1, v'_{6t}\}$ and $\{v'_1, v_{6t}\}$.

Fig. 10

A $\sigma$-tripartite labeling of $M_{18}$

It is easy to verify now that each $\ell \in [1,18t]$ occurs on exactly one edge in $M_{6t}$. Hence, the defined labeling is a $\sigma$-labeling and condition (s1) for a $\sigma$-tripartite labeling is satisfied. Condition (s2) also holds since $f(A) \subseteq [0, 6t+2]$ and $f(B \cup C) \subseteq [6t+3, 36t]$. Condition (s3) holds since $|f(v_1)-f(v_2)| + |f(v_3)-f(v_4)|=18t$, $|f(v_2)-f(v_3)| + |f(v_2)-f(v'_2)|=18t$, and $|f(v_2)-f(v'_3)|+|f(v_3)-f(v'_3)|=18t$, the number of edges of $M_{6t}$. Condition (s4) clearly holds. Also $|f(b)-f(c)|\in \{18t, 36t\}$, where $b \in B$ and $c \in C$, is impossible since $|f(b)-f(c)| \in \{1, 2, 3, 18t-1, 18t-2, 18t-3\}$. Thus, condition (s5) holds, and we have a $\sigma$-tripartite labeling of $M_{6t}$. Figure 10 shows a $\sigma$-tripartite labeling of $M_{18}$.

Case 2$n \equiv 2 \pmod {6}$.

Let $n=6t+2$ where $t \ge 2$. Thus, $|V(M_{n})|=12t+4$ and $|E(M_{n})|=18t+6$. Define a one-to-one function $f :V(M_{6t+2}) \rightarrow [0,36t+12]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t+2,&\\ f(v_{2})&= 18t+3,&\\ f(v_{3})&= 36t+8,&\\ f(v_{4})&= 18t+5,&\\ f(v_{i})&= i+2,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t+1\}, \\ f(v_{i})&= 18t-2i+10,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 6 \le i \le 2t+4\},\\ f(v_{i})&= 18t-2i+7,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t+4< i \le 6t\},\\ f(v_{6t+2})&= 18t-3,&\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t+6,&\\ f(v'_{3})&= 18t+4,&\\ f(v'_{4})&= 5,&\\ f(v'_{i})&= i+2,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 6 \le i \le 6t\},\\ f(v'_{i})&= 18t-2i+10,&v'_i \in B'_1=\{v'_i:i\ \mathrm{odd},\ 5 \le i \le 2t+3\},\\ f(v'_{i})&= 18t-2i+7,&v'_i \in B'_2=\{v'_i:i\ \mathrm{odd},\ 2t+3 < i \le 6t-1\},\\ f(v'_{6t+1})&= 18t-5,\\ f(v'_{6t+2})&= 1. \end{aligned}$$

Note that $A = \{v'_1, v'_4, v'_{6t+2}\} \cup A_1 \cup A'_1$, $B =\{v_2, v_4, v_{6t+2}, v'_3, v'_{6t+1} \} \cup B_1 \cup B_2 \cup B'_1 \cup B'_2$, and $C = \{v_1, v_3, v'_2\}$. If we proceed as in Case 1, it is easy to verify that we have a $\sigma$-tripartite labeling of $M_{6t+2}$. Figure 11 shows a $\sigma$-tripartite labeling of $M_{14}.$

Fig. 11

A $\sigma$-tripartite labeling of $M_{14}$

Case 3$n \equiv 4 \pmod {6}$.

Let $n=6t-2$, where $t \ge 3$. Thus, $|V(M_{n})|=12t-4$ and $|E(M_{n})|=18t-6$. Define a one-to-one function $f :V(M_{6t-2}) \rightarrow [0,36t-12]$ as follows:

$$\begin{aligned} f(v_{1})&= 18t-6,&\\ f(v_{2})&= 18t-7,&\\ f(v_{3})&= 36t-16,&\\ f(v_{i})&= i-1,&v_i \in A_1=\{v_i:i\ \mathrm{odd},\ 5 \le i \le 6t-3\}, \\ f(v_{i})&= 18t-2i-1,&v_i \in B_1=\{v_i:i\ \mathrm{even},\ 4 \le i \le 2t\},\\ f(v_{i})&= 18t-2i-3,&v_i \in B_2=\{v_i:i\ \mathrm{even},\ 2t< i \le 4t-2\},\\ f(v_{i})&= 18t-2i-5,&v_i \in B_3=\{v_i:i\ \mathrm{even},\ 4t-2< i \le 6t-4\},\\ f(v_{6t-2})&= 12t-2,&\\ f(v'_{1})&= 0,&\\ f(v'_{2})&= 18t-10,&\\ f(v'_{3})&= 18t-8,&\\ f(v'_{i})&= i-1,&v'_i \in A'_1=\{v'_i:i\ \mathrm{even},\ 4 \le i \le 6t-2\},\\ f(v'_{i})&= 18t-2i-1,&v'_i \in B'_1=\{v'_i:i\ \mathrm{odd},\ 5 \le i \le 2t-1\},\\ f(v'_{i})&= 18t-2i-3,&v'_i \in B'_2=\{v'_i:i\ \mathrm{odd},\ 2t-1< i \le 4t-3\},\\ f(v'_{i})&= 18t-2i-5,&v'_i \in B'_3=\{v'_i:i\ \mathrm{odd},\ 4t-3 < i \le 6t-3\}. \end{aligned}$$

Note that $A = \{v'_1\} \cup A_1 \cup A'_1$, $B =\{v_2, v_{6t-2}, v'_3\} \cup B_1 \cup B_2 \cup B_3 \cup B'_1 \cup B'_2 \cup B'_3$, and $C = \{v_1, v_3, v'_2\}$. If we proceed as in Case 1, it is easy to verify that we have a $\sigma$-tripartite labeling of $M_{6t-2}$. Figure 12 shows a $\sigma$-tripartite labeling of $M_{16}$. $\square$

Fig. 12

A $\sigma$-tripartite labeling of $M_{16}$

Because it is known that bipartite prisms and bipartite Möbius ladders admit $\alpha$-labelings and in light of our results here, we have the following.

Corollary 14

If G of size n is a prism (other than $D_3$) or a Möbius ladder, then there exists a cyclic G-decomposition of $K_{2nt+1}$, of $K_{2nt}+I$, and of $K_{2nt+2}-I$ for all positive integers t.