Proper Connection Number of Graph Products

Abstract

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographic, strong, Cartesian, and direct products and present exact values or upper bounds for these products of graphs.

1 Introduction

All graphs considered in this paper are simple, finite, and undirected. We follow the terminology and notation of Bondy and Murty [1]. For a graph G, we use V(G), E(G), $\delta (G)$, and diam(G) to denote the vertex set, edge set, minimum degree, and diameter of G, respectively. The rainbow connections of a graph which are applied to measure the safety of a network are introduced by Chartrand et al. [2]. Readers can see [2,3,4,5,6,7,8,9] for details. An edge-coloring of a graph G is an assignment c of colors to the edges of G, one color to each edge of G. Consider an edge-coloring (not necessarily proper) of a graph $G=(V,E)$. We say that a path of G is rainbow, if no two edges on the path have the same color. An edge-colored graph G is rainbow connected if every two vertices are connected by a rainbow path. The minimum number of colors required to rainbow color a graph G is called the rainbow connection number, denoted by rc(G). For more results on the rainbow connection, we refer to the survey paper [10] of Li et al. and a new book [11] of Li and Sun.

If adjacent edges of G are assigned different colors by c, then c is a proper (edge-) coloring. The minimum number of colors needed in a proper-coloring of G is referred to as the chromatic index of G and denoted by $\chi '(G)$. Recently, Andrews et al. [12] introduced the concept of proper-path colorings. Let G be an edge-colored graph, where adjacent edges may be colored the same. A path P in G is called a proper path if no two adjacent edges of P are colored the same. An edge-coloring c is a proper-path coloring of a connected graph G if every pair of distinct vertices u, v of G is connected by a proper u-v path in G. A graph with a proper-path coloring is said to be proper connected. If k colors are used, then c is referred to as a proper-path k-coloring. The minimum number of colors needed to produce a proper-path coloring of G is called the proper connection number of G, denoted by pc(G).

Let G be a nontrivial connected graph of order n and size m. Then the proper connection number of G has the following bounds.

$$\begin{aligned} 1\le pc(G)\le \min \{\chi '(G),rc(G)\}\le m. \end{aligned}$$

Furthermore, $pc(G)=1$ if and only if $G=K_n$ and $pc(G)=m$ if and only if $G = K_{1,m}$ is a star of order $m+1$. For more details on the proper connection number, we refer to [12,13,14].

The standard products (Cartesian, direct, strong, and lexicographic) draw a constant attention of graph research community, see some papers [15,16,17,18].

In this paper, we consider four standard products: the lexicographic, the strong, the Cartesian, and the direct with respect to the proper connection number. Each of these four products will be treated in one of the forthcoming sections.

2 The Cartesian Product

The Cartesian product of two graphs G and H, written as $G\Box H$, is the graph with vertex set $V(G)\times V(H)$, in which two vertices gh and $g'h'$ are adjacent if and only if $g=g'$ and $(h,h')\in E(H)$, or $h=h'$ and $(g,g')\in E(G)$. Clearly, the Cartesian product is commutative, that is, $G\Box H$ is isomorphic to $H\Box G$.

In this section G and H are two connected graphs with $V(G)=\{g_1,g_2,\ldots ,g_{n}\}$ and $V(H)=\{h_1,h_2,\ldots ,h_{m}\}$, respectively. Then $V(G\Box H)=\{g_ih_j\,|\,1\le i\le n, \ 1\le j\le m\}$. For $h\in V(H)$, we use G(h) to denote the subgraph of $G\Box H$ induced by the vertex set $\{g_ih\,|\,1\le i\le n\}$. Similarly, for $g\in V(G)$, we use H(g) to denote the subgraph of $G\Box H$ induced by the vertex set $\{gh_j\,|\,1\le j\le m\}$.

Theorem 1

Let G and H be connected graphs with $|V(G)|\ge 2$ and $|V(H)|\ge 2$. Then

$$\begin{aligned} pc(G\Box H)=2. \end{aligned}$$

Proof

Since $G\Box H$ is not a complete graph, it follows that $pc(G\Box H)\ge 2$. We need to show that $pc(G\Box H)\le 2$. To show that $pc(G\Box H)\le 2$, we provide a proper-coloring c of $G\Box H$ with two colors as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} c(gh_s,gh_t)=\mathrm{blue} &{} \mathrm{if}~h_sh_t\in E(H),\\ c(g_ih,g_jh)=\mathrm{red} &{}\mathrm{if}~g_ig_j\in E(G).\\ \end{array} \right. \end{aligned}$$

It suffices to check that there is a proper path between any two vertices $gh,g'h'$ in $G\Box H$. For any vertices $gh,gh'\in H(g)$ in the same H-layer, there is a proper path starting/ending with a blue edge and a proper path starting/ending with a red edge. If $h=h_0h_1\ldots h_{\ell }=h'$ is a $hh'$-path in H and $gg'$ is an edge in G, then the desired paths are

$$\begin{aligned} gh_0,gh_1,g'h_1,g'h_2,gh_2,\ldots ,gh' \end{aligned}$$

and

$$\begin{aligned} gh_0,g'h_0,g'h_1,gh_1,gh_2,\ldots ,gh'. \end{aligned}$$

The same is true for any two vertices in the same G-layer.

For any two vertices $gh,g'h'$ in $G\Box H$, we consider $h\ne h'$ and $g\ne g'$. Let $g=g_0g_1\ldots g_{k}=g'$. If $k\le \ell$, then the path induced by the edges in

$$\begin{aligned} g_0h_0, g_1h_0, g_1h_1, g_2h_1,\cdots , g_kh_{k-1}, g_kh_k, g_{k-1}h_k,g_{k-1}h_{k+1},\cdots , g_kh_\ell . \end{aligned}$$

is proper $gh, g'h'$-path in $G\Box H$. If $k> \ell$, then the path induced by the edges in

$$\begin{aligned} g_0h_0, g_1h_0, g_1h_1, g_2h_1,\cdots , g_\ell h_{\ell -1}, g_\ell h_\ell , g_{\ell +1}h_\ell , g_{\ell +1}h_{\ell -1},\cdots , g_kh_\ell . \end{aligned}$$

is a proper $gh, g'h'$-path in $G\Box H$. From the argument, we have $pc(G\Box H)=2$. $\square$

3 The Strong Product and Lexicographical Product

The strong product $G\boxtimes H$ of graphs G and H has the vertex set $V(G)\times V(H)$. Two vertices gh and $g'h'$ are adjacent whenever $gg'\in E(G)$ and $h=h'$, or $g=g'$ and $(h,h') \in E(H)$, or $(g,g')\in E(G)$ and $(h,h')\in E(H)$. The strong product is connected whenever both factors are and the vertex connectivity of the strong product was solved recently by Špacapan in [19]. The lexicographic product $G\circ H$ of graphs G and H has the vertex set $V(G\circ H)=V(G)\times V(H)$. Two vertices $gh,g'h'$ are adjacent if $gg'\in E(G)$, or if $g=g'$ and $hh'\in E(H)$. The lexicographic product is not commutative and is connected whenever G is connected.

Lemma 1

[12] Let G be a nontrivial connected graph, and H be a connected spanning subgraph of G. Then

$$\begin{aligned} pc(G)\le pc(H). \end{aligned}$$

The following proposition is immediate.

Proposition 1

Let G and H be nontrivial graphs. Then

$$\begin{aligned} pc(G\boxtimes H)=\left\{ \begin{array}{ll} 2 &{}\mathrm{if}~G \ and \ H \ are \ not \ both \ complete,\\ 1 &{}\mathrm{if}~G \ and \ H \ are \ complete. \end{array} \right. \end{aligned}$$

Proof

By Lemma 1 and Theorem 1, we have $pc(G\boxtimes H)\le pc(G\Box H)=2$. If G and H are both complete, then $G\boxtimes H$ is a complete graph, and hence $pc(G\boxtimes H)=1$, as desired. If G and H are not both complete, then $G\boxtimes H$ is not a complete graph, and hence $pc(G\boxtimes H)\ge 2$. So $pc(G\boxtimes H)=2$, as desired. $\square$

Since $G\boxtimes H$ is a spanning subgraph of $G\circ H$, the following result is immediate.

Proposition 2

Let G and H be nontrivial graphs. Then

$$\begin{aligned} pc(G\circ H)=\left\{ \begin{array}{ll} 2 &{}\mathrm{if}~G \ and \ H \ are \ not \ both \ complete,\\ 1 &{} \mathrm{if}~G \ and \ H \ are \ complete. \end{array} \right. \end{aligned}$$

4 The Direct Product

The direct product $G\times H$ of graphs G and H has the vertex set $V(G)\times V(H)$, and two vertices gh and $g'h'$ are adjacent if the projections on both coordinates are adjacent, i.e., $(g,g')\in E(G)$ and $(h,h')\in E(H)$. It is clearly commutative and associativity also follows quickly. For more general properties we recommend [20]. The direct product is the most natural graph product in the sense of categories. But this also seems to be the reason that it is, in general, also the most elusive product of all standard products. For example, $G\times H$ need not be connected even when both factors are. To gain connectedness of $G\times H$ at least one factor must additionally be nonbipartite as shown by Weichsel [21]. Also, the distance formula

$$\begin{aligned} d_{G\times H}(gh,g'h') = \min \{\max \{d^{e}_{G}(gg'),d^{e}_{H} (hh')\},\max \{d^{o}_{G}(gg'), d^{o}_{H}(hh')\}\} \end{aligned}$$

for the direct product is far more complicated as it is for other standard products. Here $d^{e} _{G}(g,g')$ represents the length of a shortest even walk between g and $g'$ in G, and $d^{o}_{G} (g,g')$ the length of a shortest odd walk between g and $g'$ in G. The formula was first shown in [22] and later in [23] in an equivalent version. There is no final solution for the connectivity of the direct product, only some partial results are known (see [24, 25]). But the edge connectivity of direct products is completely solved in [26].

In this section we construct different upper bounds for the proper connection number of the direct product with respect to some invariants of the factors that are related to the proper connection number of the factors. A similar concept as for the distance formula is used and is due to the proper odd and even walks between vertices (and not only proper paths) and is thus, in a way, related with the formula. We say that G is odd–even proper connected if there exists a proper colored odd path and a proper colored even path between every pair of (not necessarily different) vertices of G. The odd–even proper connection number of a graph G, oepc(G), is the smallest number of colors needed for G to be odd–even proper connected and it equals infinity if no such coloring exists. A bipartite graph has either only even or only odd paths between two fixed vertices, and thus there is no odd–even proper-coloring of such a graph. On the other hand, let G be a graph in which every edge lies on some odd cycle. Then oepc(G) is finite since coloring every edge with its own color produces an odd–even proper-coloring of G.

One can see that a odd cycle is an example where this coloring is optimal, and $oepc(G)\le |E(G)|$ for a connected graph G.

It is also easy to see that $oepc(K_{3})=3$. For $n\ge 3$, and n is odd, $oepc(C_{n})=3$. For $n\ge 3$, and n is even, $oepc(C_{n})=\infty$.

Let G be a graph. We split G into two spanning subgraphs $O^{G}$ and $B^{G}$, where the set $E(O^{G})$ consists of all edges of G that lie on some odd cycle of G, and the set $E(B^{G}) = E(G){\setminus } E(O^{G})$. Clearly, $O^{G}$ and $B^{G}$ are not always connected. Let $O^{G}_{ 1} ,O^{G}_{ 2 },\cdots ,O^{G}_{ k}$ and $B^{G}_{1},B^{G}_{2},\cdots ,B^{G}_{\ell }$ be components of $O^{G}$ and $B^{G}$, respectively, each one containing more than one vertex. Let

$$\begin{aligned} o(G)=oepc(O^{G}_{1})+oepc(O^{G}_{2})+\cdots + oepc(O^{G}_{k}), \end{aligned}$$

and

$$\begin{aligned} b(G)=pc(B^{G}_{1})+pc(B^{G}_{2})+\cdots + pc(B^{G}_{\ell }). \end{aligned}$$

Note that o(G) is finite since it is defined on nontrivial components $O^{G}_{i}$, $i\in \{1,2,\cdots ,k\}$. Set $oepc(O_i^G)=0$   if $O_i^G$ is the empty graph.

Theorem 2

Let G and H be nonbipartite connected graphs. Then

$$\begin{aligned} pc(G\times H)\le \min \{pc(H)((b(G)+o(G)), pc(G)(b(H)+o(H))\}. \end{aligned}$$

Proof

Without loss of generality, we assume that $pc(H)((b(G)+o(G))\le pc(G)(b(H)+o(H))$. Denote by $c^{B}_{G}$ an optimal proper-coloring of components of $B^{G}$. Let $c^{O}_{G}$ be an optimal odd–even proper-coloring of components of $O^{G}$ and c(H) be an optimal proper-coloring of H.

We give a proper-coloring of $G\times H$ as follows. If $e\in E(G\times H)$ projects on G to $e'\in B_G$, we set $c(e)=(c^{B} _{G}(e'),c_H(e''))$, and if e projects on G to $e'\in O_G$, we set $c(e)=(c^{O}_{G}(e'),c_H(e''))$, where $e''\in E(H)$ is the projection of e on H. By this way, we get a coloring of $V(G\times H)$ with $pc(H)(o(G)+b(G))$ colors and it remains to show that this is a proper-coloring of $G\times H$.

Let gh and $g'h'$ be arbitrary vertices from $G\times H$. Clearly, there is a proper path connecting g and $g'$, say $P=gg_{1},\ldots g_{\ell -1}g'$. There is a proper path connecting h and $h'$, say $Q=hh_{1},\ldots h_{k-1}h'$. Observe that P is a proper $g, g'$-path in G induced by ${B}_{G}$ or ${O}_{G}$ or ${B}_{G}$ and ${O}_{G}$, and Q is a proper $h, h'$-path in H. If $g=g'$ or $h=h'$, then P or Q, respectively, is a trivial one vertex path.

We distinguish the following two cases to prove this theorem.

Case 1 $\ell$ and k have the same parity.

If $h=h'$, then we let $h_{k-1}$ be an arbitrary neighbor of h. Then the path induced by the edges in

$$\begin{aligned} \{(gh, g_{1}h_{k-1}),(g_{1}h_{k-1}, g_{2}h),(g_{2}h,g_{3} h_{k-1}),\ldots ,(g_{\ell -1}h_{k-1}, g'h')\} \end{aligned}$$

is a proper $gh,g'h'$-path in $G\times H$.

If $g=g'$, then we let $g_{\ell -1}$ be an arbitrary neighbor of g. Then the path induced by the edges in

$$\begin{aligned} \{(gh, g_{\ell -1}h_{1}),(g_{\ell -1}h_{1}, gh_{2}),(gh_{2}, g_{\ell -1} h_{3}),\ldots ,(g_{\ell -1}h_{k-1}, g'h')\} \end{aligned}$$

is a proper $gh,g'h'$-path in $G\times H$.

If $g\ne g'$, and $h\ne h'$, then the path induced by the edges in

$$\begin{aligned}&\{(gh, g_{1}h_{1}), (g_{1}h_{1}, g_{2}h_{2})\ldots , \\&\quad (g_{k}h', g_{k+1}h_{k-1}),(g_{k+1}h_{k-1},g_{k+2}h')\ldots (g_{\ell -1}h_{k-1}, g'h')\} \end{aligned}$$

is a proper $gh,g'h'$-path in $G\times H$ whenever $\ell \ge k$, and the path induced by the edges in

$$\begin{aligned}&\{(gh, g_{1}h_{1}), (g_{1}h_{1}, g_{2}h_{2})\ldots , (g_{\ell -1}h_{\ell -1}, g'h_{\ell }),\\&\quad (g'h_{\ell }, g_{\ell -1}h_{\ell +1})\ldots , (g_{\ell -1}h_{k-1}, g'h')\} \end{aligned}$$

is a proper $gh,g'h'$-path in $G\times H$ whenever $\ell < k$.

Case 2 $\ell$ and k have different parity.

If there exists a $g_{i}, g_{j}$-subpath of P in $O^{G}_{p}$, we replace this subpath by a proper $g_{i}, g_{j}$-path of different parity in $O^{G}_{p}$ to obtain a proper path $P'$ between g and $g'$. If this is the case, then $|E(P')|$ and k have the same parity and we can use Case 1. We now assume that all the $g_{i}, g_{j}$-subpaths of P in $B^{G}_{p}$, that is, all vertices of P are in $B^{G}_{p}$. To find a proper $gh, g'h'$-path in $G\times H$, we find out a $g,g'$-walk in G. Note that P is contained in one component $B^{G}_{q}$. Let $g_{i}\in V(P)$ be a vertex that is closest to any component $O^{G}_{p}$ of G and let $v_{1}\in O^{G}_{p}$ be closest to $g_{i}$. Let $R = g_{i}g'_{i+1},\ldots , g'_{i+r} \ (g'_{i+r}=v_{1})$ be a shortest $g_{i}, v_{1}$-path. From the definition of odd–even proper-coloring, we know that there exists an odd proper $v_{1}, v_{1}$-cycle $C = v_{1}v_{2},\ldots v_{p}v_{1}$ in $O^{G}_{p}$. Now we insert a closed walk that follows RCR from $g_{i}$ into a path P to obtain a $g, g'$-walk

$$\begin{aligned} W= & {} gg_{1}\ldots g_{i}g'_{i+1}\ldots g'_{i+r}v_{2}v_{3},\ldots v_{p}v_{1}g'_{i+r-1}g'_{i+r-2}\ldots g'_{i+1}g_{i}g_{i+1}\ldots g'\\= & {} u_{0}u_{1},\ldots u_{\ell +p+2r} \end{aligned}$$

of length $t=\ell +2r+p$. Note that t and $\ell$ have different parity since p is an odd number, and thus t and k have the same parity. If $k\ge t$, then the path induced by the edges in

$$\begin{aligned}&\{(u_0h, u_1h_1), (u_1h_1, u_2h_2),\cdots (u_th_t, u_{t-1}h_{t+1}),\\&\quad (u_{t-1}h_{t+1}, u_th_{t+2}),\cdots (u_{t-1}h_{k-1}, u_th')\} \end{aligned}$$

is a proper-coloring connected gh and $g'h'$.

If $k<t$, then the path induced by the edges in

$$\begin{aligned}&\{(u_0h, u_1h_1), (u_1h_1, u_2h_2),\cdots (u_{k-1}h_{k-1}, u_kh'),\\&\quad (u_kh', u_{k+1}h_{k-1}),\cdots (u_{t-1}h_{k-1}, u_th')\} \end{aligned}$$

is a proper-coloring connected gh and $g'h'$. $\square$

Corollary 1

Let G and H be connected graphs, where G is nonbipartite and H is bipartite. Then

$$\begin{aligned} pc(G\times H)\le pc(H)(b(G)+o(G)). \end{aligned}$$

A bipartite graph $G=(V_{0}\cup V_{1},E)$ is said to have a property $\pi$ if G admits of an automorphism $\psi$ such that $x\in V_{0}$ if and only if $\psi (x)\in V_{1}$. For more details, we refer to [27].

Lemma 2

[27] If G and H are bipartite graphs one of which has property $\pi$, then the two components of $G\times H$ are isomorphic.

Proposition 3

Let G be a nonbipartite connected graph. Then

$$\begin{aligned} pc(G\times K_{2})\le o(G)+b(G). \end{aligned}$$

Proof

Let $c^{O}_{G}$ be an optimal odd–even proper-coloring of $O^{_{G}}$ and let $c^{B}_{G}$ be an optimal proper-coloring of $B^{_{G}}$ (for both cases it holds that no color appears in two different components). Observe that $c^{O}_{G}=o(G)$ and $c^{B}_{G}=b(G)$. We provide a coloring c of $G\times K_{2}$ with $o(G)+b(G)$ colors as follows.

Recall that $O^{G}_{1} ,O^{G}_{2},\cdots ,O^{G}_{k}$ and $B^{G}_{1},B^{G}_{2},\cdots ,B^{G}_{\ell }$ are all the components of $O^{G}$ and $B^{G}$, respectively. By the definition, $B^{G}_{i}$ is bipartite graph. From Lemma 2, $B^{G}_{i}\times K_{2}$ can be decomposed into two subgraphs isomorphic to $B^{G}_{i}$. Color both components of $B^{G}_{i}\times K_{2}$ (which are isomorphic to $B^{G}_{i}$) optimally with $pc(B^{G}_{i})$ colors for every $i\in \{1,2,\ldots , \ell \}$. For this we use b(G) colors. Now, we assign o(G) new colors to the remaining edges. For an edge $(gh, g'h')$ of $G\times K_{2}$, it projects on G to an edge $gg'$ of $O^{G}$ receive color $c(gh, g'h')=c^{O}_{G}(gg')$. For an edge $(gh, g'h')$ of $G\times K_{2}$, it projects on G to an edge $gg'$ of $B^{G}$ receive color $c(gh, g'h')=c^{B}_{G}(gg')$. For the introduced coloring $o(G)+b(G)$ colors are used and we need to show that c is a proper-coloring of $G\times K_{2}$.

Set $V(K_{2})= \{k_1, k_2\}$. Let gh and $g'h'$ be arbitrary vertices in $G\times K_{2}$. Let $P=gg_{1},\ldots g_{\ell -1}g'$ be a proper $g,g'$-path under the proper-coloring of G induced by $c^{O}_{G}$ and $c^{B}_{G}$. We distinguish two cases to show this proposition.

Case 1 Let $\ell$ and $d_{K_{2}}(h,h')$ have the same parity.

Without loss of generality we may assume that $h=k_{1}$. Consequently $h'= k_{1}$ if $\ell$ is an even number and $h'=k_{2}$ otherwise. Thus,

$$\begin{aligned} (gk_{1})(g_{1}k_{2})(g_{2}k_{1})\cdots (g'h') \end{aligned}$$

is a proper $(g,h),(g',h')$-path in $G\times K_{2}$.

Case 2 Let $\ell$ and $d_{K_{2}}(h,h')$ have different parity.

Suppose first that P has a nonempty intersection with some $O^{G} _{p}$ and let $g_{i}$ be the first and $g_{j}$ the last vertex of P in $O^{G}_{p}$. Then we can find a proper $g_{i},g_{j}$-path in $O^{G}_{p}$ with length of different parity as is the length of the $g_{i},g_{j}$-subpath of P in $O^{G}_{p}$. Replacing the $g_{i},g_{j}$-subpath of P by this proper $g_{i},g_{j}$-path in $O^{G}_{p}$, we obtain a proper $g,g'$-path of the same parity as $d_{K_{2}}(h,h')$ and we continue as in Case 1.

Suppose now that P has an empty intersection with every $O^{G}_{p}$, $p\in \{1,2,\ldots ,k\}$. Then P is contained in $B^{G}_{q}$ for some q, and gh and $g'h'$ are in different components $(B^{G}_{ q})_{1}$ and $(B^{G}_{q})_{2}$ of $B^{G}_{q}\times K_{2}$, respectively. Since G is nonbipartite, there exists a vertex $g''$ in some component of $O^{G}_{p}$. Set $\{h_{r},h_{s}\}=\{k_{1},k_{2}\}$. Take a proper path from gh to $g''h_{r}$ in $(B^{G}_{q} )_{1}$, a proper odd path from $g''h_{r}$ to $g''h_{s}$ in $O^{G}_{p}$ , and a rainbow path from $g''h_{s}$ to $g'h'$ in $(B^{G}_{q})_{2}$. This is a proper $gh,g'h'$-path in $G\times K_{2}$. $\square$

5 Applications

In this section, we demonstrate the usefulness of the proposed constructions by applying them to some instances of Cartesian and lexicographic product networks. From Propositions 1, 2, and Theorem 1, any network that is a nontrivial Cartesian product has proper connection number 2, and any network that is a strong or lexicographic product has proper connection number 1 or 2.

Fig. 1

a Petersen graph; b the network $HP_4$; c the structure of $HL_4$.

A r-dimensional mesh is the Cartesian product of r linear arrays $P_{m_1},P_{m_2},\cdots ,P_{m_r}$. A r-dimensional hypercube is a special case of a r-dimensional mesh, in which the r linear arrays are all of size 2; see [28]. A r-dimensional torus is the Cartesian product of r cycles $C_{m_1},C_{m_2},\cdots ,C_{m_r}$ of size at least three. The cycles $C_{m_i}$ are not necessary to have the same size. Let $K_n$ be a clique of n vertices, $n\ge 2$. An r dimensional generalized hypercube [29, 30] is the Cartesian product of r cliques. The network $P_{m_1}\circ P_{m_2}\circ \cdots \circ P_{m_r}$, $C_{m_1}\circ C_{m_2}\circ \cdots \circ C_{m_r}$ and $K_{m_1}\circ K_{m_2}\circ \cdots \circ K_{m_r}$ are investigated in [31].

• For r-dimensional mesh $P_{m_1}\Box P_{m_2}\Box \cdots \Box P_{m_r}$,

  $$\begin{aligned} pc(P_{m_1}\Box P_{m_2}\Box \cdots \Box P_{m_r})=2. \end{aligned}$$

• For network $C_{m_1}\Box C_{m_2}\Box \cdots \Box C_{m_r}$,

  $$\begin{aligned} pc(C_{m_1}\Box C_{m_2}\Box \cdots \Box C_{m_r})=2. \end{aligned}$$

• For network $K_{m_1}\Box K_{m_2}\Box \cdots \Box K_{m_r} \ (m_i\ge 2, \ r\ge 2, \ 1\le i\le r)$,

  $$\begin{aligned} pc(K_{m_1}\Box K_{m_2}\Box \cdots \Box K_{m_r})=2. \end{aligned}$$

• For network $P_{m_1}\circ P_{m_2}\circ \cdots \circ P_{m_r}$,

  $$\begin{aligned} pc(P_{m_1}\circ P_{m_2}\circ \cdots \circ P_{m_r})=\left\{ \begin{array}{ll} 2 &{} \mathrm{if}~\mathrm{there}~\mathrm{exists}~\mathrm{some}~m_j~\mathrm{such}~\mathrm{that}~m_j\ne 2,\\ 1 &{}\mathrm{if}~m_1=m_2=\cdots =m_r=2, \end{array} \right. \end{aligned}$$

  where $m_i$ is the order of $P_{m_i}$ and $1\le i\le r$.

• For network $C_{m_1}\circ C_{m_2}\circ \cdots \circ C_{m_r}$,

  $$\begin{aligned} pc(C_{m_1}\circ C_{m_2}\circ \cdots \circ C_{m_r})=\left\{ \begin{array}{ll} 2 &{} \mathrm{if}~\mathrm{there}~\mathrm{exists}~\mathrm{some}~m_j~\mathrm{such}~\mathrm{that}~m_j\ge 4,\\ [0.2cm] 1 &{}\mathrm{if}~m_1=m_2=\cdots =m_r=3, \end{array} \right. \end{aligned}$$

  where $m_i$ is the order of $C_{m_i}$ and $1\le i\le r$.

• For network $K_{m_1}\circ K_{m_2}\circ \cdots \circ K_{m_r}$,

  $$\begin{aligned} pc(K_{m_1}\circ K_{m_2}\circ \cdots \circ K_{m_r})=1. \end{aligned}$$

A r-dimensional hyper Petersen network $HP_r$ is the Cartesian product of $Q_{r-3}$ and the well-known Petersen graph [32], where $r\ge 3$ and $Q_{r-3}$ denotes an $(n-3)$-dimensional hypercube. The cases $r=3$ and 4 of hyper Petersen networks are depicted in Fig. 1. Note that $HP_3$ is just the Petersen graph; see Fig. 1a.

The network $HL_r$ is the lexicographic product of $Q_{r-3}$ and the Petersen graph, where $r\ge 3$ and $Q_{r-3}$ denotes an $(r-3)$-dimensional hypercube; see [31]. Note that $HL_3$ is just the Petersen graph, and $HL_4$ is a graph obtained from two copies of the Petersen graph by adding one edge between one vertex in a copy of the Petersen graph and one vertex in another copy. See Fig. 1c for an example (We only show the edges $v_1u_i \ (1\le i\le 10)$).

• For network $HP_3$ and $HL_3$, $pc(HP_3)=pc(HL_3)=2$;

• For network $HL_4$ and $HP_4$, $pc(HP_4)=pc(HL_4)=2$.