On the Coalition Number of the dth Power of the n-Cycle

Abstract

A coalition in a graph G consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1\cup V_2$ is a dominating set. A coalition partition in a graph G is a vertex partition $\pi =\{V_1,V_2,\dots ,V_k\}$ such that every set $V_i\in \pi$ is not a dominating set but forms a coalition with another set $V_j\in \pi$ which is not a dominating set. The coalition number $C\left(G\right)$ equals the maximum k of a coalition partition of G. In this paper, we study the coalition number of the dth power of the n-cycle $C_n^d$, where $n\ge 3$ and $d\ge 2$. We show that $C\left(C_n^d\right)=d^2+3d+2$ for $n=2d^2+4d+2$ or $n\ge 2d^2+5d+3$, and also provide some bounds of $C\left(C_n^d\right)$ for the other cases. As a special case, we obtain the exact value of the coalition number of $C_n^2$.

1. Introduction

All graphs considered in this paper are finite and simple. Let $G=(V,E)$ be a graph with vertex set V and edge set E. The open neighborhood of a vertex $v\in V$ is the set $N\left(v\right)=\left\{u\right|uv\in E\}$, and its closed neighborhood is the set $N\left[v\right]=\left\{v\right\}\cup N\left(v\right)$. Each vertex $u\in N\left(v\right)$ is called a neighbor of v, and $\left|N\right(v\left)\right|$ is the degree of v, denoted $deg\left(v\right)$. In a graph G of order n, a vertex of degree $n-1$ is called a full or universal vertex, while a vertex of degree 0 is an isolate vertex. The minimum degree of G is denoted by $\delta \left(G\right)$ and the maximum degree by $\Delta \left(G\right)$. A subset $V_i\subseteq V$ is called a singleton set if $|V_i|=1$ or a non-singleton set if $|V_i|\ge 2$. A set $S\subseteq V$ is a dominating set of a graph G if every vertex in $V-S$ is adjacent to at least one vertex in S. For a set $S\subseteq V$, its open neighborhood is the set $N\left(S\right)={\cup }_{v\in S}N\left(v\right)$, and its closed neighborhood is the set $N\left[S\right]=N\left(S\right)\cup S$.

For two positive integers a and b with $a\le b$, we denote by $[a,b]$ the interval of integers $\{a,\dots ,b\}$. Note that $[a,b]=\varnothing$ when $a>b$. By $\diamond [a,b]$ ($\circ [a,b]$), we denote the set of all even (odd) numbers in the interval $[a,b]$.

Coalitions and coalition partitions were introduced by Haynes et al. [1] in 2020, where they defined the concepts in terms of general graph properties but focused on the property of being a dominating set. After reading the paper [1], we immediately thought that this could be a very good research direction. We know that in graph theory, the study of domination and related topics is one of the fascinating and fast-developing areas. So far, more than 2000 research papers on domination have been published in various journals.

A coalition in a graph G consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set of G but whose union $V_1\cup V_2$ is a dominating set of G. We say that the sets $V_1$ and $V_2$ form a coalition and that they are coalition partners. A coalition partition, henceforth called a c-partition in a graph G is a vertex partition $\pi =\{V_1,V_2,\dots ,V_k\}$ such that every set $V_i$ of $\pi$ is either a dominating set of G, or is not a dominating set of G but forms a coalition with another non-dominating set $V_j\in \pi$. The coalition number $C\left(G\right)$ equals the maximum order k of a c-partition of G. The singleton partition of a graph G with vertex set $\{v_1,v_2,\dots ,v_n\}$ is the partition ${\pi }_1=\{V_1,V_2,\dots ,V_n\}$, where $V_i=\left\{v_i\right\}$ for $i\in [1,n]$. To aid in discussion, we sometimes assign to each vertex a color $i\in [1,k]$ and define the partition as $\{V_1,V_2,\dots ,V_k\}$, where $V_i$ is the set of vertices colored i.

In [1], Haynes et al. provided an example to illustrate these definitions. Consider the path $P_6=(v_1,v_2,v_3,v_4,v_5,v_6)$. Let $V_1=\{v_1,v_6\}$ and $V_i=\left\{v_i\right\}$ for $i\in [2,5]$. See Figure 1. The partition $\pi =\{V_1,V_2,V_3,V_4,V_5\}$ is a c-partition of $P_6$. No set of $\pi$ is a dominating set, but $V_1$ and $V_3$ form a coalition; $V_1$ and $V_4$ form a coalition; $V_2$ and $V_5$ form a coalition. Thus, every set forms a coalition with at least one other set, and $C\left(P_6\right)\ge 5$ follows. The only partition of $V\left(P_6\right)$ of larger order is the singleton partition ${\pi }_1=\{V_1,V_2,V_3,V_4,V_5,V_6\}$, where $V_i=\left\{v_i\right\}$ for $i\in [1,6]$. Since no dominating set of $P_6$ contains $v_1$ and one other vertex, i.e., the set $V_1$ does not form a coalition with any other set in ${\pi }_1$, and therefore, the singleton partition ${\pi }_1$ of $P_6$ is not a c-partition. Hence, $C\left(P_6\right)=5$ and $\pi =\{V_1,V_2,V_3,V_4,V_5\}$ is a $C\left(P_6\right)$-partition of $P_6$.

Let G be a graph with a c-partition $\pi =\{V_1,V_2,\dots ,V_k\}$. The coalition graph $CG(G,\pi )$ of G is the graph with vertex set $\{V_1,V_2,\dots ,V_k\}$, corresponding one-to-one with the sets of $\pi$, and two vertices $V_i$ and $V_j$ are adjacent in $CG(G,\pi )$ if and only if the sets $V_i$ and $V_j$ are coalition partners in $\pi$, that is, neither $V_i$ nor $V_j$ is a dominating set of G, but $V_i\cup V_j$ is a dominating set of G. A graph G is called a self-coalition graph if G is isomorphic to its coalition graph $CG(G,{\pi }_1)$, where ${\pi }_1$ is the singleton partition of G.

In 2020, Haynes et al. [1] first introduced the concepts of coalition, coalition partition, and coalition number of a graph G, studied these properties, and provided some bounds about $C\left(G\right)$. In addition, they determined the exact values of $C\left(P_n\right)$ and $C\left(C_n\right)$ for $n\ge 3$, where $P_n$ and $C_n$ denote the path and cycle with n vertices, respectively. In 2021, Haynes et al. [2] showed that $C\left(G\right)\le {(\Delta \left(G\right)+3)}^2/4$ for any graph G and $C\left(G\right)\le \left(\delta \right(G)+1)(\Delta (G)-\delta (G)+2)$ for a graph G with no full vertices and $\delta \left(G\right)<\Delta \left(G\right)/2$. In 2023, Davood Bakhshesh et al. [3] characterized all graphs G of order n with $\delta \left(G\right)\le 1$ and $C\left(G\right)=n$. Moreover, they characterized all trees T of order n with $C\left(T\right)=n-1$. In 2024, Saeid Alikhani et al. [4] studied the coalition number of cubic graphs and provided the exact value of the coalition number of cubic graphs with at most 10 vertices. In the same year, Dobrynin and Golmohammadi [5] constructed an infinite family of cubic graphs satisfying $C\left(G\right)=9$.

In 2023, Haynes et al. [6] first studied the coalition graph and showed that every graph is a coalition graph. Then, they [7] studied and characterized self-coalition graphs. Shortly after, they [8] studied and characterized the coalition graphs of paths, cycles, and trees. In the same year, Davood Bakhshesh et al. [3] determined the number of coalition graphs that can be defined by all coalition partitions of a given path. In 2024, Dobrynin and Golmohammadi [9] showed that $C_{15}$ is the shortest cycle having the maximal number of coalition graphs.

After reading some papers, we found that it is not as simple as we thought, and we plan to start with some simple questions. We would like to extend these results related to $C\left(P_n\right)$ and $C\left(C_n\right)$ provided by Haynes et al. [6]. Firstly, we tried to study the power of a path and the power of a cycle. To our surprise, we obtained some results about the coalition number of these graphs. Then, we decided to study the general case. We studied the coalition numbers of the dth power of the n-path $P_n$, denoted $P_n^d$, obtained the exact values of $C\left(P_n^d\right)$ for a sufficiently large n, and also provided some bounds of $C\left(P_n^d\right)$ for the other cases. As a special case, we obtained exact values of $C\left(P_n^2\right)$ except for $n\in [11,20]$. See [10].

In this paper, we investigate the coalition number of the dth power of the n-cycle $C_n$, denoted $C_n^d$, where $n\ge 3$ and $d\ge 2$. In Section 2, we provide the exact values or bounds of the coalition number of $C_n^d$. In Section 3, we determine the exact value of the coalition number of $C_n^2$.

2. The dth Power of the $\mathbf{n}$-Cycle

For any positive integers $n\ge 3$ and $d\ge 2$, the dth power of the n-cycle $C_n$, denoted $C_n^d$, has the vertex set

$$V\left(C_n^d\right)=\{v_1,v_2,\dots ,v_n\}$$

and the edge set

$$E\left(C_n^d\right)=\bigcup _{1\le i\le n}\left\{v_i{v}_j\right|j=i+1,i+2,\dots ,i+d\},$$

where the index j for $v_j$ is taken as modulo n. For $n\le 2d+1$, $C_n^d\cong K_n$ and $C\left(C_n^d\right)=C\left(K_n\right)=n$. Thus, in this paper, we only need to consider $C_n^d$ for $n\ge 2d+2$.

Lemma 1 

([1]). Let G be a graph with maximum degree $\Delta \left(G\right)$, and let π be a $C\left(G\right)$-partition. If $X\in \pi$, then X is in, at most, $\Delta \left(G\right)+1$ coalitions.

Lemma 2 

([2]). For any graph G with maximum degree $\Delta \left(G\right)$,

$$C\left(G\right)\le {\displaystyle \frac{{(\Delta \left(G\right)+3)}^2}{4}}.$$

Since $\Delta \left(C_n^d\right)=2d$, it is easy to see that if $n\in [2d+2,4d+2]$, then for each vertex $x\in V\left(C_n^d\right)$, there exists another vertex $y\in V\left(C_n^d\right)$, such that $\{x,y\}$ is a dominating set of $C_n^d$; that is, $\left\{x\right\}$ and $\left\{y\right\}$ form a coalition of $C_n^d$. Thus, we have the following result.

Proposition 1. 

For any $d\ge 2$, if $n\in [2d+2,4d+2]$, then we have

$$C\left(C_n^d\right)=n.$$

Proposition 2. 

For any $d\ge 2$, if $n\in [4d+3,5d+2]$, then we have

$$n-d\le C\left(C_n^d\right)\le n-2.$$

Proof. 

Suppose that $n\in [4d+3,5d+2]$. Since for each vertex $x\in V\left(C_n^d\right)$, we have $\left|N\right[x\left]\right|=2d+1$, then any two vertices of $C_n^d$ cannot be a dominating set. Thus, we have $C\left(C_n^d\right)\le n-1$. Suppose that $C\left(C_n^d\right)=n-1$ and $\pi =\{V_1,V_2,\dots ,V_{n-1}\}$ is a $C\left(C_n^d\right)$-partition of $C_n^d$. Then, there is only one set in $\pi$ with two vertices, and the other sets in $\pi$ are all singleton sets. Without the loss of generality, we may assume that $|V_1|=2$ and $|V_i|=1$ for each $i\in [2,n-1]$. By Lemma 1, $V_1$ can form a coalition with at most $2d+1$ singleton sets, respectively. So there are at least $(n-1)-1-(2d+1)\ge (4d+3-1)-1-(2d+1)=2d$ singleton sets in $\pi$ such that each of them cannot form a coalition with another set in $\pi$, a contradiction to the fact that $\pi =\{V_1,V_2,\dots ,V_{n-1}\}$ is a $C\left(C_n^d\right)$–partition of $C_n^d$. Thus, we have,

$$C\left(C_n^d\right)\le n-2.$$

(1)

Now we define a labeling $\ell :V\left(C_n^d\right)\to [1,n-d]$ of the vertices of $C_n^d$ as follows. Let,

$$\ell \left(v_i\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{for}i\in [1,2d+2];\hfill \\ i-2d-1,\hfill & \mathrm{for}i\in [2d+3,3d+1];\hfill \\ 1,\hfill & \mathrm{for}i=3d+2;\hfill \\ i-d,\hfill & \mathrm{for}i\in [3d+3,n].\hfill \end{array}\right.$$

Let $V_j=\left\{v_i\right|\ell \left(v_i\right)=j,i\in [1,n]\}$ for $j\in [1,n-d]$. Then, we have $V_1=\{v_1,v_{3d+2}\}$, $V_j=\{v_j,v_{j+2d+1}\}$ for $j\in [2,d]$, $V_j=\left\{v_j\right\}$ for $j\in [d+1,2d+2]$, and $V_j=\left\{v_{j+d}\right\}$ for $j\in [2d+3,n-d]$. See Figure 2 for an example. It is easy to check that, for each $j\in [d+1,2d+2]$, $V_1$ and $V_j$ form a coalition; for each $j\in [2d+i+1,min\{3d+i+2,n-d\left\}\right]$, $V_i$ and $V_j$ form a coalition, where $i\in [2,d]$. This means that $\{V_1,V_2,\dots ,V_{n-d}\}$ is a c-partition. So we have

$$C\left(C_n^d\right)\ge n-d.$$

(2)

Combining inequalities (1) and (2), we have $n-d\le C\left(C_n^d\right)\le n-2$. □

Proposition 3. 

For any $d\ge 2$, if $n\in [5d+3,6d+4]$, then we have

$$C\left(C_n^d\right)\ge 4d+2.$$

Proof. 

Suppose that $n\in [5d+3,6d+4]$. Now we consider three cases.

Case 1. $n=5d+3$.

We define a labeling $\ell :V\left(C_n^d\right)\to [1,4d+2]$ as follows. Let

$$\ell \left(v_i\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{for}i\in [1,2d+2];\hfill \\ i-2d-1,\hfill & \mathrm{for}i\in [2d+3,3d+2];\hfill \\ 1,\hfill & \mathrm{for}i=3d+3;\hfill \\ i-d-1,\hfill & \mathrm{for}i\in [3d+4,5d+3].\hfill \end{array}\right.$$

Let $V_j=\left\{v_i\right|\ell \left(v_i\right)=j,i\in [1,n]\}$ for $j\in [1,4d+2]$. Then, we have $V_1=\{v_1,v_{3d+3}\}$, $V_j=\{v_j,v_{j+2d+1}\}$ for $j\in [2,d+1]$, $V_j=\left\{v_j\right\}$ for $j\in [d+2,2d+2]$, and $V_j=\left\{v_{j+d+1}\right\}$ for $j\in [2d+3,4d+2]$. See Figure 3 for an example. It is easy to check that, for each $j\in [d+2,2d+2]$, $V_1$ and $V_j$ form a coalition; for each $j\in [2d+i+1,3d+i+1]$, $V_i$ and $V_j$ form a coalition, where $i\in [2,d+1]$. This means that $\{V_1,V_2,\dots ,V_{4d+2}\}$ is a c-partition. So we have

$$C\left(C_n^d\right)\ge 4d+2.$$

Case 2. $n=5d+4$.

We define a labeling $\ell :V\left(C_n^d\right)\to [1,4d+2]$ as follows. Let

$$\ell \left(v_i\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{for}i\in [1,2d+2];\hfill \\ i-2d-1,\hfill & \mathrm{for}i\in [2d+3,3d+3];\hfill \\ 1,\hfill & \mathrm{for}i=3d+4;\hfill \\ i-d-2,\hfill & \mathrm{for}i\in [3d+5,5d+4].\hfill \end{array}\right.$$

Let $V_j=\left\{v_i\right|\ell \left(v_i\right)=j,i\in [1,n]\}$ for $j\in [1,4d+2]$. Then, we have $V_1=\{v_1,v_{3d+4}\}$, $V_j=\{v_j,v_{j+2d+1}\}$ for $j\in [2,d+2]$, $V_j=\left\{v_j\right\}$ for $j\in [d+3,2d+2]$, and $V_j=\left\{v_{j+d+2}\right\}$ for $j\in [2d+3,4d+2]$. See Figure 4 for an example. It is easy to check that, for each $j\in [d+3,2d+2]$, $V_1$ and $V_j$ form a coalition; for each $j\in [2d+i+1,3d+i]$, $V_i$ and $V_j$ form a coalition, where $i\in [2,d+2]$. This means that $\{V_1,V_2,\dots ,V_{4d+2}\}$ is a c-partition. So we have

$$C\left(C_n^d\right)\ge 4d+2.$$

Case 3. $n\in [5d+5,6d+4]$.

We define a labeling $\ell :V\left(C_n^d\right)\to [1,4d+2]$ as follows. Let

$$\ell \left(v_i\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{for}i\in [1,2d+2];\hfill \\ i-2d-1,\hfill & \mathrm{for}i\in [2d+3,3d+2];\hfill \\ 1,\hfill & \mathrm{for}i=3d+3;\hfill \\ i-d-1,\hfill & \mathrm{for}i\in [3d+4,5d+3];\hfill \\ i-5d-3,\hfill & \mathrm{for}i\in [5d+4,n].\hfill \end{array}\right.$$

Let $V_j=\left\{v_i\right|\ell \left(v_i\right)=j,i\in [1,n]\}$ for $j\in [1,4d+2]$. Then, we have $V_1=\{v_1,v_{3d+3},v_{5d+4}\}$, $V_j=\{v_j,v_{j+2d+1},v_{j+5d+3}\}$ for $j\in [2,n-5d-3]$, $V_j=\{v_j,v_{j+2d+1}\}$ for $j\in [n-5d-2,d+1]$, $V_j=\left\{v_j\right\}$ for $j\in [d+2,2d+2]$, and $V_j=\left\{v_{j+d+1}\right\}$ for $j\in [2d+3,4d+2]$. See Figure 5 for an example. It is easy to check that, for each $j\in [d+2,2d+2]$, $V_1$ and $V_j$ form a coalition; for each $j\in [2d+i+1,3d+i+1]$, $V_j$ and $V_i$ form a coalition, where $i\in [2,n-5d-3]$; for each $j\in [2d+i+3,3d+i+1]$, $V_j$ and $V_i$ form a coalition, where $i\in [n-5d-2,d+1]$. This means that $\{V_1,V_2,\dots ,V_{4d+2}\}$ is a c-partition. So we have

$$C\left(C_n^d\right)\ge 4d+2.$$

Combining Cases 1–3, we have $C\left(C_n^d\right)\ge 4d+2$ for $n\in [5d+3,6d+4]$. □

Proposition 4. 

For any $d\ge 2$, if $n=2d^2+4d+2$ or $n\ge 2d^2+5d+3$, then we have

$$C\left(C_n^d\right)=d^2+3d+2.$$

Proof. 

Suppose that $n=2d^2+4d+2$ or $n\ge 2d^2+5d+3$. Since $\Delta \left(C_n^d\right)=2d$, then by Lemma 2 we have

$$C\left(C_n^d\right)\le {\displaystyle \frac{{(\Delta \left(C_n^d\right)+3)}^2}{4}}={\displaystyle \frac{{(2d+3)}^2}{4}}={\displaystyle \frac{4d^2+12d+9}{4}},$$

and then

$$C\left(C_n^d\right)\le d^2+3d+2.$$

(3)

In order to show that $C\left(C_n^d\right)=d^2+3d+2$, in the following, we label the vertices of the graph $C_n^d$ with integers $i\in [1,d^2+3d+2]$, define the set $V_i$ to consist of all vertices labeled i, and then show that $\pi =\{V_1,V_2,\dots ,V_{d^2+3d+2}\}$ is a c-partition of $C_n^d$.

Let $S_i=\{v_{(i-1)d+i},v_{(i-1)d+i+1},\dots ,v_{min\{id+i,n\}}\}$, in which $i\in [1,\lceil {\displaystyle \frac{n}{d+1}}\rceil ]$. Define a labeling $\ell :V\left(C_n^d\right)\to [1,d^2+3d+2]$ of the vertices of the graph $C_n^d$ as follows, let

$$\ell \left(v_j\right)=\left\{\begin{array}{cc}{\ell }^{\prime }\left(v_j\right),\hfill & \mathrm{if}{\ell }^{\prime }\left(v_j\right)\le d+1\mathrm{for}{v}_j\in S_i,i\in \circ [1,2d+2];\hfill \\ {\ell }^{\prime }\left(v_j\right)-(d+1),\hfill & \mathrm{if}{\ell }^{\prime }\left(v_j\right)\ge d+2\mathrm{for}{v}_j\in S_i,i\in \circ [1,2d+2];\hfill \\ {\ell }^{\prime }\left(v_j\right),\hfill & \mathrm{for}{v}_j\in S_i,i\in \diamond [1,2d+2];\hfill \\ {\ell }^{\prime }\left(v_j\right),\hfill & \mathrm{for}{v}_j\in S_i,i\in [2d+3,\lceil {\displaystyle \frac{n}{d+1}}\rceil ],\hfill \end{array}\right.$$

where

$${\ell }^{\prime }\left(v_j\right)=\left\{\begin{array}{cc}j-(i-1)d-{\displaystyle \frac{(i-1)}{2}},\hfill & \mathrm{for}{v}_j\in S_i,i\in \circ [1,2d+2];\hfill \\ j-({\displaystyle \frac{i}{2}}-1)(d+1),\hfill & \mathrm{for}{v}_j\in S_i,i\in \diamond [1,2d+2];\hfill \\ j-(i-1)d-(i-1),\hfill & \mathrm{for}{v}_j\in S_i,i\in [2d+3,\lceil {\displaystyle \frac{n}{d+1}}\rceil ].\hfill \end{array}\right.$$

It is easy to see that $\ell \left(v_j\right)\in [1,d+1]$ for any $v_j\in S_i$, where $i\in \circ [1,2d+2]\cup [2d+3,\lceil {\displaystyle \frac{n}{d+1}}\rceil ]$ and $\ell \left(v_j\right)\in [{\displaystyle \frac{i}{2}}d+({\displaystyle \frac{i}{2}}+1),({\displaystyle \frac{i}{2}}+1)d+({\displaystyle \frac{i}{2}}+1)]$ for any $v_j\in S_i$, where $i\in \diamond [1,2d+2]$. Thus, we have $\ell \left(v_j\right)\in [1,d^2+3d+2]$ for each $j\in [1,n]$. For any $k\in [1,d^2+3d+2]$, let $V_k=\left\{v_j\right|\ell \left(v_j\right)=k,j\in [1,n]\}$. See Figure 6 for an example. Note that for each $k\in [1,d^2+3d+2]$, $V_k$ is not a dominating set of $C_n^d$, and it is not difficult to check that for each $i\in [1,d+1]$, $V_i$ forms a coalition with $V_j$, where $j\in [id+(i+1),(i+1)d+(i+1\left)\right]$. Then, $\pi =\{V_1,V_2,\dots ,V_{d^2+3d+2}\}$ is a c-partition of $C_n^d$. Then, we have

$$C\left(C_n^d\right)\ge d^2+3d+2.$$

(4)

Combining inequalities (3) and (4), we have $C\left(C_n^d\right)=d^2+3d+2$. □

For $n=2d^2+4d+1$ or $2d^2+5d+2$, by the above proof, we know that the subset $\{V_1,V_2,\dots ,V_{d^2+3d+1}\}$ of $\pi =\{V_1,V_2,\dots ,V_{d^2+3d+2}\}$, defined in the proof of Proposition 4, is a c-partition of $C_n^d$. Then, we have the following result.

Corollary 1. 

For any $d\ge 2$, if $n=2d^2+4d+1$ or $2d^2+5d+2$, then we have

$$C\left(C_n^d\right)\ge d^2+3d+1.$$

Until now, the results of $C\left(C_n^d\right)$ can be summarized in

Table 1. Values of $C\left(C_n^d\right)$, where $n\ge 3$ and $d\ge 2$.

n	$[3,4d+2]$	$[4d+3,5d+2]$	$[5d+3,6d+4]$	$[6d+5,2d^2+4d]$	$2d^2+4d+1$
$C\left(C_n^d\right)$	n	$[n-d,n-2]$	$[4d+2,d^2+3d+2]$	$\le d^2+3d+2$	$[d^2+3d+1,d^2+3d+2]$
	$2d^2+4d+2$	$[2d^2+4d+3,2d^2+5d+1]$	$2d^2+5d+2$	$\ge 2d^2+5d+3$
	$d^2+3d+2$	$\le d^2+3d+2$	$[d^2+3d+1,d^2+3d+2]$	$d^2+3d+2$

.

For the fixed value of $d=2,3$, we have the relations between $C\left(C_n^d\right)$ and n as follows (

Table 2. Values of $C\left(C_n^d\right)$, where $n\ge 3$ and $d=2,3$.

n	$[3,10]$	$[11,12]$	$[13,16]$	17	18	19	20	$\ge 21$
$C\left(C_n^2\right)$	n	$n-2$	$[10,12]$	$[11,12]$	12	$\le 12$	$[11,12]$	12
n	$[3,14]$	$[15,17]$	$[18,22]$	$[23,30]$	31	32	$[33,34]$	35	$\ge 36$
$C\left(C_n^3\right)$	n	$[n-3,n-2]$	$[14,20]$	$\le 20$	$[19,20]$	20	$\le 20$	$[19,20]$	20

).

It is easy to see that the larger the value of d, the wider the range where $C_n^d$ cannot be determined. Determining the exact values of $C_n^d$ may be difficult.

Proposition 5. 

Let $\pi =\{V_1,V_2,\dots ,V_k\}$ be a c-partition of $C_n^d$ and $n\equiv i(mod2d+1)$, where $d\ge 2$ and $n>2d+1$. For a set $V_{\ell }\in \pi$ with $\lceil {\displaystyle \frac{n}{2d+1}}\rceil -1$ vertices, if $i=0$, then $V_{\ell }$ can form a coalition with at most one singleton set in π; if $i\in [1,2d]$, then $V_{\ell }$ can form a coalition with at most $2d-i+2$ singleton sets in π.

Proof. 

Suppose that $d\ge 2$, $n>2d+1$, and $n\equiv i(mod2d+1)$. Suppose that $\pi =\{V_1,V_2,\dots ,V_k\}$ a c-partition of $C_n^d$ includes a set $V_{\ell }$ with $\lceil {\displaystyle \frac{n}{2d+1}}\rceil -1$ vertices. If no singleton set exists in $\pi$, or there exist singleton sets in $\pi$, but any one of these cannot form a coalition with $V_{\ell }$, then there is nothing to prove. Suppose that there exists a singleton set in $\pi$ forming a coalition with $V_{\ell }$. It is not difficult to see that the vertices in $N\left[V_{\ell }\right]$ are consecutive. Otherwise, the singleton set that forms a coalition with $V_{\ell }$ cannot dominate the vertices in $V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]$, which is a contradiction. Since the vertices in $V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]$ can be dominated by a singleton set in $\pi$, the vertices in $V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]$ must be consecutive. Without loss of generality, we may assume that $V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]=\{v_1,v_2,\dots ,v_s\}$. Now we consider the following two cases.

Case 1. $i=0$.

Since there exists a singleton set in $\pi$ forming a coalition with $V_{\ell }$, we have $|V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]|=2d+1$ and only $v_{d+1}$ can dominate the vertices in $V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]$, i.e., only $\left\{v_{d+1}\right\}$ can form a coalition with $V_{\ell }$.

Case 2. $i\in [1,2d]$.

Since there exists a singleton set in $\pi$ forming a coalition with $V_{\ell }$, we have $i\le |V\left(C_n^d\right)\setminus N\left[V_{\ell }\right]|\le 2d+1$, i.e., $i\le s\le 2d+1$. It is easy to see that each vertex in $\{v_{s-d},\dots ,v_{d+1}\}$ can dominate the set $\{v_1,v_2,\dots ,v_s\}$, i.e., each vertex in $\{v_{s-d},\dots ,v_{d+1}\}$ as a singleton set can form a coalition with $V_{\ell }$. Since $i\le s\le 2d+1$, then we have $s-d\le d+1$ and $|\{v_{s-d},\dots ,v_{d+1}\}|=(d+1)-(s-d)+1=2d-s+2\le 2d-i+2$.

The proof is complete. □

3. The Power of the n-Cycle

As a special case, now we consider $C\left(C_n^d\right)$ for $d=2$ and obtain the exact values of $C\left(C_n^2\right)$ as follows. For convenience, without loss of generality, we may assume that $|V_1|\ge |V_2|\ge \cdots \ge |V_k|$ for a c-partition $\pi =\{V_1,V_2,\dots ,V_k\}$ of a graph G. By Proposition 5, as $d=2$ we have the following result.

Corollary 2. 

Let $\pi =\{V_1,V_2,\dots ,V_k\}$ be a c-partition of $C_n^2$ and $n\equiv i(mod5)$, where $n\ge 6$. For a set $V_{\ell }\in \pi$ with $\lceil {\displaystyle \frac{n}{5}}\rceil -1$ vertices, if $i=0$, then $V_{\ell }$ can form a coalition with at most one singleton set in π; if $i\in [1,4]$, then $V_{\ell }$ can form a coalition with at most $6-i$ singleton sets in π.

Corollary 3. 

For any $n\ge 3$, we have

$$C\left(C_n^2\right)=\left\{\begin{array}{cc}n,\hfill & n\in [3,10];\hfill \\ 9,\hfill & n=11;\hfill \\ 10,\hfill & n\in [12,16];\hfill \\ 11,\hfill & n\in \{17,19,20\};\hfill \\ 12,\hfill & n=18\mathrm{or}n\ge 21.\hfill \end{array}\right.$$

Proof. 

It is easy to see that, for $n\in [3,5]$, $C_n^2\cong K_n$ and $C\left(C_n^2\right)=C\left(K_n\right)=n$. Suppose that $n\ge 6$ is a positive integer. By Proposition 1, we have $C\left(C_n^2\right)=n$ for $n\in [6,10]$. By Proposition 2, we have $C\left(C_n^2\right)=n-2$ for $n\in [11,12]$. By Proposition 4, we have $C\left(C_n^2\right)=12$ for $n=18$ or $n\ge 21$. By Proposition 3, we have $C\left(C_n^2\right)\ge 10$ for $n\in [13,16]$. By Corollary 1, we have $C\left(C_n^2\right)\ge 11$ for $n\in \{17,20\}$. In the following, we only need to show that $C\left(C_n^2\right)\le 10$ for $n\in [13,16]$, $C\left(C_n^2\right)\le 11$ for $n\in \{17,20\}$, and $C\left(C_{19}^2\right)=11$.

Claim A. 

$C\left(C_n^2\right)\le 10$ for $n\in [13,16]$.

Proof of Claim A. 

By Lemma 2, $C\left(C_n^2\right)\le 12$ for any $n\ge 6$. Now we show that $C\left(C_n^2\right)\le 10$ for $n\in [13,16]$ by contradiction. Now we consider the following four cases.

Case 1. $n=13$.

Suppose that $C\left(C_{13}^2\right)=12$ and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{13}^2$. Then we just need to consider the case: $|V_1|=2$ and $|V_2|=|V_3|=\cdots =|V_{12}|=1$. By Corollary 2, $V_1$ can form a coalition with each of, at most, 3 singleton sets in $\pi$. Then, in $\pi$, there are at least eight singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction. Therefore, we have $C\left(C_{13}^2\right)\le 11$.

Suppose that $C\left(C_{13}^2\right)=11$ and $\pi =\{V_1,V_2,\dots ,V_{11}\}$ is a c-partition of $C_{13}^2$. Then, we need to consider the two cases as follows.

Subcase 1.1. $|V_1|=3$ and $|V_2|=|V_3|=\cdots =|V_{11}|=1$.

By Lemma 1, $V_1$ can form a coalition with each of, at most, five singleton sets in $\pi$. Then, in $\pi$, there are at least five singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 1.2. $|V_1|=|V_2|=2$ and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

By Corollary 2, $V_1$ ($V_2$, resp.) can form a coalition with each of, at most, three singleton sets in $\pi$. Then, in $\pi$, there are at least three singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Combining Subcases 1.1 and 1.2, we have $C\left(C_{13}^2\right)\le 10$.

Case 2. $n=14$.

Suppose that $C\left(C_{14}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{14}^2$. Then, we need to consider the two cases as follows.

Subcase 2.1. $|V_1|=3$ and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 2.2. $|V_1|=|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.2.

Combining Subcases 2.1 and 2.2, we have $C\left(C_{14}^2\right)\le 11$.

Suppose that $C\left(C_{14}^2\right)=11$, and $\pi =\{V_1,V_2,\dots ,V_{11}\}$ is a c-partition of $C_{14}^2$. Then, we need to consider the three cases as follows.

Subcase 2.3. $|V_1|=4$, and $|V_2|=|V_3|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 2.4. $|V_1|=3,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

By Lemma 1, $V_1$ can form a coalition with each of, at most, five singleton sets in $\pi$. By Corollary 2, $V_2$ can form a coalition with each of, at most, two singleton sets in $\pi$. Then, in $\pi$, there are at least two singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 2.5. $|V_1|=|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 1.2.

Combining Subcases 2.3–2.5, we have $C\left(C_{14}^2\right)\le 10$.

Case 3. $n=15$.

Suppose that $C\left(C_{15}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{15}^2$. Then, we need to consider the three cases as follows.

Subcase 3.1. $|V_1|=4$, and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 3.2. $|V_1|=3,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.4.

Subcase 3.3. $|V_1|=|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.2.

Combining Subcases 3.1–3.3, we have $C\left(C_{15}^2\right)\le 11$.

Suppose that $C\left(C_{15}^2\right)=11$, and $\pi =\{V_1,V_2,\dots ,V_{11}\}$ is a c-partition of $C_{15}^2$. Then, we need to consider the five cases as follows.

Subcase 3.4. $|V_1|=5$, and $|V_2|=|V_3|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 3.5. $|V_1|=4,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 2.4.

Subcase 3.6. $|V_1|=|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

If $V_1$ and $V_2$ form a coalition, then by Lemma 1, $V_1$ ($V_2$, resp.) can form a coalition with each of, at most, four singleton sets in $\pi$. Therefore, in $\pi$, there exist at least one singleton set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ do not form a coalition, then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_2\right]$. Thus, each set in $\pi$ forms a coalition with $V_1$ or $V_2$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$, then the number of the sets in $\pi$ that each can form a coalition with $V_1$ or $V_2$ is, at most, five, and there are at least five singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 3.7. $|V_1|=3,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 2.4.

Subcase 3.8. $|V_1|=|V_2|=|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 1.2.

Combining Subcases 3.4–3.8, we have $C\left(C_{15}^2\right)\le 10$.

Case 4. $n=16$.

Suppose that $C\left(C_{16}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{16}^2$. Then, we need to consider the five cases as follows.

Subcase 4.1. $|V_1|=5$, and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 4.2. $|V_1|=4,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

By Lemma 1, $V_1$ can form a coalition, with each of, at most, five singleton sets in $\pi$. $V_2$ cannot form a coalition with any singleton set in $\pi$. Then, in $\pi$, there are, at least, five singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 4.3. $|V_1|=|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 3.6.

Subcase 4.4. $|V_1|=3,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2.

Subcase 4.5. $|V_1|=|V_2|=\cdots =|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

It is not difficult to see that, in this case, each singleton set in $\pi$ cannot form a coalition with any other set in $\pi$, a contradiction.

Combining Subcases 4.1–4.5, we have $C\left(C_{16}^2\right)\le 11$.

Suppose that $C\left(C_{16}^2\right)=11$, and $\pi =\{V_1,V_2,\dots ,V_{11}\}$ is a c-partition of $C_{16}^2$. Then, we need to consider the seven cases as follows.

Subcase 4.6. $|V_1|=6$, and $|V_2|=|V_3|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 1.1.

Subcase 4.7. $|V_1|=5,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 4.2.

Subcase 4.8. $|V_1|=4,|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 3.6.

Subcase 4.9. $|V_1|=4,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 4.2.

Subcase 4.10. $|V_1|=|V_2|=3,|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{11}|=1$.

If $V_1$ and $V_2$ form a coalition, then by Lemma 1, $V_1$ ($V_2$, resp.) can form a coalition with each of, at most, four sets in $\pi \setminus \{V_1,V_2\}$. Then, in $\pi$, there exists at least one set that cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ do not form a coalition, then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_2\right]$. Then, each set in $\pi$ forming a coalition with $V_1$ or $V_2$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$, the number of sets in $\pi$—each of which can form a coalition with $V_1$ or $V_2$—is at most five, and at least three singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 4.11. $|V_1|=3,|V_2|=|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 4.2.

Subcase 4.12. $|V_1|=|V_2|=\cdots =|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{11}|=1$.

The proof is similar to the proof of Subcase 4.5.

Combining Subcases 4.6–4.12, we have $C\left(C_{16}^2\right)\le 10$.

Combining Cases 1–4, we have $C\left(C_n^2\right)\le 10$ for $n\in [13,16]$.

The proof of Claim A is complete. □

Claim B. 

$C\left(C_n^2\right)\le 11$ for $n\in \{17,20\}$.

Proof of Claim B. 

We consider the two cases as follows.

Case 1. $n=17$.

Suppose that $C\left(C_{17}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{17}^2$. Then, we need to consider the seven subcases as follows.

Subcase 1.1. $|V_1|=6$, and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1 in Claim A.

Subcase 1.2. $|V_1|=5,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 1.3. $|V_1|=4,|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.4 in Claim A.

Subcase 1.4. $|V_1|=4,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 1.5. $|V_1|=|V_2|=3,|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

By Corollary 2, $V_1$ ($V_2$, resp.) can form a coalition with each of, at most, four singleton sets in $\pi$. $V_3$ cannot form a coalition with any singleton set in $\pi$. Then, in $\pi$, there is, at least, one singleton set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 1.6. $|V_1|=3,|V_2|=|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Subcase 1.7. $|V_1|=|V_2|=\cdots =|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.5 in Claim A.

Combining Subcases 1.1–1.7, we have $C\left(C_{17}^2\right)\le 11$.

Case 2. $n=20$.

Suppose that $C\left(C_{20}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{20}^2$. Then, we need to consider the 22 subcases as follows.

Subcase 2.1. $|V_1|=9$, and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1 in Claim A.

Subcase 2.2. $|V_1|=8,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 2.3. $|V_1|=7,|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.4 in Claim A.

Subcase 2.4. $|V_1|=7,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 2.5. $|V_1|=6,|V_2|=4$ and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 3.6 in Claim A.

Subcase 2.6. $|V_1|=6,|V_2|=3,|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

By Lemma 1, $V_1$ can form a coalition with each of, at most, five singleton sets in $\pi$. By Corollary 2, $V_2$ can form a coalition with, at most, one singleton set in $\pi$. Since $V_3$ cannot form a coalition with any singleton set in $\pi$. Then, in $\pi$, there are, at least, three singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 2.7. $|V_1|=6,|V_2|=|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 2.8. $|V_1|=|V_2|=5$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 3.6 in Claim A.

Subcase 2.9. $|V_1|=5,|V_2|=4,|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.10 in Claim A.

Subcase 2.10. $|V_1|=5,|V_2|=|V_3|=3$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.4 in Claim A.

Subcase 2.11. $|V_1|=5,|V_2|=3,|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.6 in Claim B.

Subcase 2.12. $|V_1|=5,|V_2|=|V_3|=\cdots =|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 2.13. $|V_1|=|V_2|=4,|V_3|=3$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

If any two of the three sets $V_1,V_2$, and $V_3$ can form a coalition; then, by Lemma 1, $V_1$ ($V_2$, resp.) in $\pi$ can form a coalition with each of, at most, three singleton sets in $\pi$. By Corollary 2, $V_3$ can form a coalition with, at most, one singleton set in $\pi$. Then, in $\pi$, there are at least two singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ form a coalition, $V_1$ and $V_3$ (or $V_2$ and $V_3$) do not form a coalition. Then, there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_3\right]$. Thus, each set in $\pi$ forming a coalition with $V_1$ or $V_3$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$, the number of sets in $\pi$—each of which can form a coalition with $V_1$ or $V_3$—is at most five. Since $V_1$ and $V_2$ form a coalition, then $V_2\cap N\left[v_i\right]\ne \varnothing$. Then, the number of singleton sets in $\pi$ that each of them can form a coalition with $V_1$ or $V_3$ is, at most, four. By Lemma 1, $V_2$ can form a coalition with each of, at most, four singleton sets in $\pi$. Then, in $\pi$, there exists, at least, one set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ do not form a coalition, then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_2\right]$. Then, each set in $\pi$ forming a coalition with $V_1$ or $V_2$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$. By Corollary 2, $V_3$ can form a coalition with, at most, one singleton set in $\pi$. Then, in $\pi$, there are at least three singleton sets in which each of them cannot form a coalition with any other set in $\pi$, a contradiction.

Subcase 2.14. $|V_1|=|V_2|=4,|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

If $V_1$ and $V_2$ form a coalition, and there exist $V_i\in \{V_1,V_2\}$ and $V_j\in \{V_3,V_4\}$, such that $V_i$ and $V_j$ form a coalition. Without loss of generality, we may assume that $V_1$ and $V_3$ form a coalition. Then, by Lemma 1, $V_1$ can form a coalition with each of, at most, three singleton sets in $\pi$, and $V_2$ can form a coalition with each of, at most, four singleton sets in $\pi$. Since $V_3$ ($V_4$, resp.) cannot form a coalition with any singleton set in $\pi$. Then, in $\pi$, there exists at least one set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ form a coalition, and for each $V_i\in \{V_1,V_2\}$ and $V_j\in \{V_3,V_4\}$, $V_i$ and $V_j$ cannot form a coalition. Since $V_3$ ($V_4$, resp.) cannot form a coalition with any singleton set in $\pi$, then $V_3$ and $V_4$ must form a coalition. Note that $V_1$ and $V_3$ do not form a coalition. Then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_3\right]$. Thus, each set in $\pi$ forming a coalition with $V_1$ or $V_3$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$, the number of sets in $\pi$, each of which can form a coalition with $V_1$ or $V_3$, is at most five. Since $V_1$ and $V_2$ form a coalition, then $V_2\cap N\left[v_i\right]\ne \varnothing$. Since $V_3$ and $V_4$ form a coalition, then $V_4\cap N\left[v_i\right]\ne \varnothing$. Then, the number of singleton sets in $\pi$ that each can form a coalition with $V_1$ or $V_3$ is, at most, three. By Lemma 1, $V_2$ can form a coalition with each of, at most, four singleton sets in $\pi$. Then, in $\pi$, there exists at least one set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ do not form a coalition, then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_2\right]$. Then, each set in $\pi$ forming a coalition with $V_1$ or $V_2$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$, the number of singleton sets in $\pi$ that each of them can form a coalition with $V_1$ or $V_2$ is, at most, five, and at least three singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Subcase 2.15. $|V_1|=4,|V_2|=|V_3|=3,|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.6 in Claim B.

Subcase 2.16. $|V_1|=4,|V_2|=3,|V_3|=|V_4|=|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.6 in Claim B.

Subcase 2.17. $|V_1|=4,|V_2|=|V_3|=\cdots =|V_6|=2$, and $|V_7|=|V_8|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Subcase 2.18. $|V_1|=|V_2|=\cdots =|V_4|=3$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.2 in Claim A.

Subcase 2.19. $|V_1|=|V_2|=|V_3|=3,|V_4|=|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Subcase 2.20. $|V_1|=|V_2|=3,|V_3|=|V_4|=\cdots =|V_6|=2$, and $|V_7|=|V_8|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Subcase 2.21. $|V_1|=3,|V_2|=|V_3|=\cdots =|V_7|=2$, and $|V_8|=|V_9|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Subcase 2.22. $|V_1|=|V_2|=\cdots =|V_8|=2$, and $|V_9|=|V_{10}|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.5 in Claim A.

Combining Subcases 2.1–2.22, we have $C\left(C_{20}^2\right)\le 11$.

Combining Cases 1–2, we have $C\left(C_n^2\right)\le 11$ for $n\in \{17,20\}$.

The proof of Claim B is complete. □

Claim C. 

$C\left(C_{19}^2\right)=11$.

Proof of Claim C. 

Let $\pi =\{V_1,V_2,\dots ,V_{11}\}$, in which $V_1=\{v_1,v_9,v_{14},v_{16}\}$, $V_2=\{v_2,v_7,v_{15},v_{19}\}$, $V_3=\{v_3,v_8,v_{13}\}$, $V_4=\left\{v_4\right\}$, $V_5=\left\{v_5\right\}$, $V_6=\left\{v_6\right\}$, $V_7=\left\{v_{10}\right\}$, $V_8=\left\{v_{11}\right\}$, $V_9=\left\{v_{12}\right\}$, $V_{10}=\left\{v_{17}\right\}$, $V_{11}=\left\{v_{18}\right\}$. It is easy to check that $V_1$ can form a coalition with $V_4$, $V_5$, and $V_6$; $V_2$ can form a coalition with $V_7$, $V_8$, and $V_9$; and $V_3$ can form a coalition with $V_{10}$ and $V_{11}$. Now we have $C\left(C_{19}^2\right)\ge 11$.

Now we show that $C\left(C_{19}^2\right)=11$ by contradiction. Suppose that $C\left(C_{19}^2\right)=12$, and $\pi =\{V_1,V_2,\dots ,V_{12}\}$ is a c-partition of $C_{19}^2$. Then, we need to consider the 15 cases as follows.

Case 1. $|V_1|=8$, and $|V_2|=|V_3|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.1 in Claim A.

Case 2. $|V_1|=7,|V_2|=2$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Case 3. $|V_1|=6,|V_2|=3$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.4 in Claim A.

Case 4. $|V_1|=6,|V_2|=|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Case 5. $|V_1|=5,|V_2|=4$, and $|V_3|=|V_4|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 3.6 in Claim A.

Case 6. $|V_1|=5,|V_2|=3$$|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.6 in Claim B.

Case 7. $|V_1|=5,|V_2|=|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Case 8. $|V_1|=|V_2|=4,|V_3|=2$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.10 in Claim A.

Case 9. $|V_1|=4,|V_2|=|V_3|=3$, and $|V_4|=|V_5|=\cdots =|V_{12}|=1$.

If $V_1$ and $V_2$ form a coalition, then by Lemma 1, $V_1$ can form a coalition with each of, at most, four singleton sets in $\pi$. By Corollary 2, $V_2$ ($V_3$, resp.) can form a coalition with each of, at most, two singleton sets in $\pi$. Then, in $\pi$, there exists at least one set which cannot form a coalition with any other set in $\pi$, which is a contradiction.

If $V_1$ and $V_2$ do not form a coalition, then there exists a vertex $v_i\notin N\left[V_1\right]\cup N\left[V_2\right]$. So each set in $\pi$ forming a coalition with $V_1$ or $V_2$ must dominate $v_i$. Since $\left|N\right[v_i\left]\right|=5$. By Corollary 2, $V_3$ can form a coalition with, at most, two singleton sets in $\pi$. Then, in $\pi$, there are at least two singleton sets in which each of them cannot form a coalition with any other set in $\pi$, which is a contradiction.

Case 10. $|V_1|=4,|V_2|=3,|V_3|=|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 2.6 in Claim B.

Case 11. $|V_1|=4,|V_2|=|V_3|=\cdots =|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.2 in Claim A.

Case 12. $|V_1|=|V_2|=|V_3|=3,|V_4|=2$, and $|V_5|=|V_6|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Case 13. $|V_1|=|V_2|=3,|V_3|=|V_4|=|V_5|=2$, and $|V_6|=|V_7|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Case 14. $|V_1|=3,|V_2|=|V_3|=\cdots =|V_6|=2$, and $|V_7|=|V_8|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 1.5 in Claim B.

Case 15. $|V_1|=|V_2|=\cdots =|V_7|=2$, and $|V_8|=|V_9|=\cdots =|V_{12}|=1$.

The proof is similar to the proof of Subcase 4.5 in Claim A.

Combining Cases 1–15, we have $C\left(C_{19}^2\right)\le 11$.

The proof of Claim C is complete. □

Combining Claims A, B, and C, the proof of the main result is complete. □

4. Conclusions

In this paper, we studied the coalition number of $C_n^d$, which showed that

(1)

$C\left(C_n^d\right)=n$ for $n\in [2d+2,4d+2]$;

(2)

$n-d\le C\left(C_n^d\right)\le n-2$ for $n\in [4d+3,5d+2]$;

(3)

$C\left(C_n^d\right)\ge 4d-2$ for $n\in [5d+3,6d+4]$;

(4)

$C\left(C_n^d\right)=d^2+3d+2$ for $n=2d^2+4d+2$ or $n\ge 2d^2+5d+3$; and

(5)

$C\left(C_n^d\right)\ge d^2+3d+1$ for $n=2d^2+4d+1$ or $2d^2+5d+2$.

However, for $n\in [6d+5,2d^2+4d]\cup [2d^2+4d+3,2d^2+5d+1]$, we did not get any better lower bounds of $C\left(C_n^d\right)$. For a special case, we obtain the exact value of $C\left(C_n^2\right)$ for any $n\ge 3$. Thus, much hard work still needs to done in the future.

Since $C_n$ and $C_n^d$ are all regular graphs, for further research, it would be interesting to study the coalition number of regular graphs and other graphs with special structures. On the other hand, we have to admit that the method used to prove our main results is a bit tedious; thus, a proof with combinatorial techniques is very much expected.