Strong rainbow connection numbers of toroidal meshes

TL;DR

This paper improves the upper bound on the strong rainbow connection number of toroidal meshes, providing a tighter estimate and resolving an open problem posed by Li et al. in 2011.

Contribution

The authors significantly refine the upper bound for the strong rainbow connection number of toroidal meshes, offering a negative answer to a previously open problem.

Findings

Improved upper bound on the strong rainbow connection number

Negative resolution to Li et al.'s open problem

Enhanced understanding of rainbow connectivity in toroidal meshes

Abstract

In 2011, Li et al. \cite{LLL} obtained an upper bound of the strong rainbow connection number of an $r$-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.

Tables

Table 1. Table 1:   A path from u 𝑢 u to v 𝑣 v in C 2 ​ k ​ □ ​ C 2 subscript 𝐶 2 𝑘 □ subscript 𝐶 2 C_{2k}\square C_{2}

$u$	$v$	Condition	A path from $u$ to $v$
$is$	$js$	$1\le j-i\le k$	$(is,\left(i+1\right)s,\dots ,js)$
$is$	$js$	$j-i\ge k+1$	$(js,\left(j+1\right)s,\dots ,\left(2k-1\right)s,0s,1s,\dots ,is)$
$is$	$js$	$1\le i-j\le k$	$(js,\left(j+1\right)s,\dots ,is)$
$is$	$js$	$i-j\ge k+1$	$(is,\left(i+1\right)s,\dots ,\left(2k-1\right)s,0s,1s,\dots ,js)$
$i0$	$j1$	$1\le j-i\le k$	$(i0,i1,\left(i+1\right)1,\dots ,j1)$
$i0$	$j1$	$j-i\ge k+1$	$(j0,j1,\left(j+1\right)1,\dots ,\left(2k-1\right)1,01,11,\dots ,i1)$
$i0$	$j1$	$1\le i-j\le k$	$(j0,j1,\left(j+1\right)1,\dots ,i1)$
$i0$	$j1$	$i-j\ge k+1$	$(i0,i1,\left(i+1\right)1,\dots ,\left(2k-1\right)1,01,11,\dots ,j1)$

Table 2. Table 2:   A path from u 𝑢 u to v 𝑣 v in C 2 ​ k + 1 ​ □ ​ C 2 subscript 𝐶 2 𝑘 1 □ subscript 𝐶 2 C_{2k+1}\square C_{2}

$u$	$v$	Condition	A path from $u$ to $v$
$is$	$js$	$1\le j-i\le k$	$(is,\left(i+1\right)s,\dots ,js)$
$is$	$js$	$j-i\ge k+1$	$(js,\left(j+1\right)s,\dots ,\left(2k\right)s,0s,1s,\dots ,is)$
$is$	$js$	$1\le i-j\le k$	$(js,\left(j+1\right)s,\dots ,is)$
$is$	$js$	$i-j\ge k+1$	$(is,\left(i+1\right)s,\dots ,\left(2k\right)s,0s,1s,\dots ,js)$
$00$	$j1$	$0\le j\le k$	$(00,01,11,\dots ,j1)$
$00$	$j1$	$k+1\le j\le 2k$	$(00,01,\left(2k\right)1,\dots ,j1)$
$10$	$j1$	$1\le j\le k+1$	$(10,20,\dots ,j0,j1)$
$10$	$j1$	$j=0$ or $k+2\le j\le 2k$	$(10,00,01,\left(2k\right)1,\dots ,j1)$
$i0$	$j1$	$2\le i\le k-1$ and $0\le j\le i$	$(i0,i1,\left(i-1\right)1,\dots ,j1)$
$i0$	$j1$	$2\le i\le k-1$ and $i+1\le j\le i+k$	$(i0,\left(i+1\right)0,\dots ,j0,j1)$
$i0$	$j1$	$2\le i\le k-1$ and $i+k+1\le j\le 2k$	$(i0,\left(i-1\right)0,\dots ,00,01,\left(2k\right)1,\left(2k-1\right)1,\dots ,j1)$
$k0$	$j1$	$0\le j\le k$	$(k0,k1,\left(k-1\right)1,\dots ,j1)$
$k0$	$j1$	$k+1\le j\le 2k$	$(k0,\left(k+1\right)0,\dots ,j0,j1)$
$i0$	$j1$	$k+1\le i\le 2k-1$ and $0\le j\le i-k-1$	$(i0,\left(i+1\right)0,\dots ,00,01,\dots ,j1)$
$i0$	$j1$	$k+1\le i\le 2k-1$ and $i-k\le j\le i$	$(i0,i1,\left(i-1\right)1,\dots ,j1)$
$i0$	$j1$	$k+1\le i\le 2k-1$ and $i+1\le j\le 2k$	$(i0,\left(i+1\right)0,\dots ,j0,j1)$
$\left(2k\right)0$	$j1$	$0\le j\le k-1$	$(\left(2k\right)0,00,01,\dots ,j1)$
$\left(2k\right)0$	$j1$	$k\le j\le 2k$	$(\left(2k\right)0,\left(2k\right)1,\left(2k-1\right)1,\dots ,j1)$

Equations

$$\sum _{1\le k\le r}\lfloor \frac{n_k}{2}\rfloor \le \mathrm{rc}(C_{n_1}\square \cdots \square C_{n_r})\le \mathrm{src}(C_{n_1}\square \cdots \square C_{n_r})\le \sum _{1\le k\le r}\lceil \frac{n_k}{2}\rceil .$$

$$\mathrm{rc}(C_{n_1}\square \cdots \square C_{n_r})=\mathrm{src}(C_{n_1}\square \cdots \square C_{n_r})=\sum _{1\le k\le r}\frac{n_k}{2}.$$

$$\mathrm{src}(C(G,S))=\mathrm{rc}(C(G,S))=\sum _{a\in S^{\ast }}\lceil \frac{|a|}{2}\rceil ?$$

$${\rm src}(C_{n_{1}}\square\cdots\square C_{n_{r}})\leq\left\{\begin{array}[]{lll}\left\lceil\dfrac{n_{1}+\cdots+n_{r}-\mu}{2}\right\rceil,&\hbox{$0\leq\mu\leq\left\lfloor\dfrac{r}{2}\right\rfloor$};\\ \\ \left\lceil\dfrac{n_{1}+\cdots+n_{r}-r+\mu}{2}\right\rceil,&\hbox{$\left\lfloor\dfrac{r}{2}\right\rfloor+1\leq\mu\leq r$,}\end{array}\right.$$

$$\mathrm{src}(\Gamma )\le \mathrm{src}(\Gamma \square \Lambda )\le \mathrm{src}(\Gamma )+\mathrm{src}(\Lambda ).$$

$$f_{1}(e)=\left\{\begin{array}[]{lll}k,&\hbox{if $e=\{i0,i1\}$};\\ i,&\hbox{if $e=\{ij,(i+1)j\},0\leq i\leq k-1$};\\ i-k,&\hbox{if $e=\{ij,(i+1)j\}$, $k\leq i\leq 2k-2$};\\ k-1,&\hbox{if $e=\{(2k-1)j,0j\}$}.\end{array}\right.$$

$$f_{2}(e)=\left\{\begin{array}[]{lll}i,&\hbox{if $e=\{i0,i1\},1\leq i\leq k$};\\ i-k-1,&\hbox{if $e=\{i0,i1\},k+1\leq i\leq 2k$};\\ k,&\hbox{if $e=\{00,01\}$};\\ i,&\hbox{if $e=\{ij,(i+1)j\}$, $0\leq i\leq k$};\\ i-k-1,&\hbox{if $e=\{ij,(i+1)j\}$, $k+1\leq i\leq 2k-1$};\\ k-1,&\hbox{if $e=\{(2k)j,0j\}$}.\end{array}\right.$$

$$\mathrm{src}(\Gamma \square C_n)\le \lceil \frac{n-2}{2}\rceil +\mathrm{src}(\Gamma \square C_2),n\ge 3.$$

$$\mathrm{src}(\Gamma \square C_3)\le \mathrm{src}(C_3)+\mathrm{src}(\Gamma )=1+\mathrm{src}(\Gamma )\le 1+\mathrm{src}(\Gamma \square C_2).$$

$$f_{3}(e)=\left\{\begin{array}[]{lll}i,&\hbox{if $e\in E[\Gamma x_{i},\Gamma x_{i+1}]$},\hbox{$1\leq i\leq\left\lfloor\frac{n}{2}\right\rfloor-1$};\\ i-\left\lfloor\frac{n}{2}\right\rfloor,&\hbox{if $e\in E[\Gamma x_{i},\Gamma x_{i+1}]$},\hbox{$\left\lfloor\frac{n}{2}\right\rfloor+1\leq i\leq n-1$};\\ f_{2,i}(e),&\hbox{if $e\in E(L_{i})$ for $1\leq i\leq 2$};\\ f_{1}(\{y_{1},y_{2}\}),&\hbox{if $e=\{y_{1}x_{i},y_{2}x_{i}\}\in E[\Gamma x_{i},\Gamma x_{i}]$, $n\geq 5$ and $i\in I$},\end{array}\right.$$

$$(u=y_1{x}_i,y_1{x}_{i+1},\dots ,y_1{x}_j)\cup P_1$$

$$(u=y_1{x}_i,y_1{x}_{i+1},\dots ,y_1{x}_{\lfloor \frac{n}{2}\rfloor })\cup P_2\cup (y_2{x}_{\lfloor \frac{n}{2}\rfloor +1},y_2{x}_{\lfloor \frac{n}{2}\rfloor +2},\dots ,y_2{x}_j=v)$$

$$(u=y_1{x}_i,y_1{x}_{i-1},\dots ,y_1{x}_1)\cup P_3\cup (y_2{x}_n,y_2{x}_{n-1},\dots ,y_2{x}_{j+1},y_2{x}_j=v)$$

$$\mathrm{src}(C_{n_1}\square \cdots \square C_{n_r})\le \lceil \frac{n_1+\cdots +n_r}{2}\rceil .$$

$$\mathrm{src}(C_{n_1}\square C_{n_2})$$

$$\mathrm{src}(C_{n_1}\square \cdots \square C_{n_{r-1}}\square C_{n_r})$$

$$\mathrm{src}(C_{n_1}\square \cdots \square C_{n_r})$$

$$\mathrm{src}(C_{n_1}\square C_{n_2})\le \lceil \frac{n_2-2}{2}\rceil +\mathrm{src}(C_{n_1}\square C_2)=\lceil \frac{n_2-2}{2}\rceil +\lceil \frac{n_1+1}{2}\rceil .$$

$$\mathrm{src}(C_{n_1}\square C_{n_2})=\mathrm{src}(C_{n_2}\square C_{n_1})\le \lceil \frac{n_1-2}{2}\rceil +\mathrm{src}(C_{n_2}\square C_2)=\lceil \frac{n_1+n_2-1}{2}\rceil .$$

$$\mathrm{src}(C_{n_1}\square C_{n_2}\square C_{n_3})\le \mathrm{src}(C_{n_1})+\mathrm{src}(C_{n_2}\square C_{n_3})\le \lceil \frac{n_1+n_2+n_3-1}{2}\rceil .$$

$$\mathrm{src}(C_{n_1}\square \cdots \square C_{n_r})$$

Full text

Key words: toroidal mesh; (strong) rainbow path; (strong) rainbow connection number; Cayley graph.

1 Introduction

All graphs considered in this paper are finite, connected and simple. We refer to the book [2] for graph theory notation and terminology not described here. Let $\Gamma$ be a graph. Denote by $V(\Gamma)$ and $E(\Gamma)$ the vertex set and edge set of $\Gamma$, respectively. A sequence of distinct vertices is a path if any two consecutive vertices are adjacent. A path $P:(v_{1},v_{2},\ldots,v_{k})$ is a cycle if $v_{1}$ is adjacent to $v_{k}$, denoted by $C_{k}$. The distance, $d(u,v)$, between vertices $u$ and $v$ is equal to the length of a shortest path connecting $u$ and $v$. The diameter of $\Gamma$, $d(\Gamma)$, is the maximum distance between two vertices in $\Gamma$ over all pairs of vertices.

Define a $k$-edge-coloring $\zeta:E(\Gamma)\rightarrow\{1,2,\ldots,k\}$, $k\in\mathbb{N}$, where adjacent edges may be colored the same. A path is rainbow if no two edges of it are colored the same. A path from $u$ to $v$ is called a strong rainbow path if it’s a rainbow path with length $d(u,v)$. If any two distinct vertices $u$ and $v$ of $\Gamma$ are connected by a (strong) rainbow path, then $\Gamma$ is called (strong) rainbow-connected under the coloring $\zeta$, and $\zeta$ is called a (strong) rainbow $k$-coloring of $\Gamma$. The (strong) rainbow connection number of $\Gamma$, denoted by (src$(\Gamma)$) rc$(\Gamma)$, is the minimum $k$ for which there exists a (strong) rainbow $k$-coloring of $\Gamma$. Clearly, we have $d(\Gamma)\leq$ rc$(\Gamma)\leq$ src$(\Gamma)$.

The (strong) rainbow connection number of a graph was first introduced by Chartrand et al. [3]. Ananth and Nasre [1] proved that, for every integer $k\geq 3$, deciding whether ${\rm src}(\Gamma)\leq k$ is NP-hard even when $\Gamma$ is bipartite. (Strong) rainbow connection numbers of some special graphs have been studied in the literature, such as outerplanar graphs [4], Cayley graphs [5, 8], line graphs [6], power graphs [7], undirected double-loop networks [9] and non-commuting graphs [10].

The Cartesian product of two graphs $\Gamma$ and $\Lambda$ is the graph $\Gamma\square\Lambda$ whose vertex set is the set $\{\gamma\lambda|\gamma\in V(\Gamma),\lambda\in V(\Lambda)\}$, and two vertices $\gamma\lambda,\gamma^{\prime}\lambda^{\prime}$ are adjacent if $\lambda=\lambda^{\prime}$ and $\{\gamma,\gamma^{\prime}\}\in E(\Gamma)$ or if $\gamma=\gamma^{\prime}$ and $\{\lambda,\lambda^{\prime}\}\in E(\Lambda)$. The Cartesian product operation is commutative and associative, hence the Cartesian product of more factors is well-defined. The graph $C_{n_{1}}\square\cdots\square C_{n_{r}}$ is an $r$-dimensional undirected toroidal mesh, where $n_{k}\geq 2$ for $1\leq k\leq r$.

In 2011, Li et al. proved the following theorem and proposed an open problem.

Theorem 1.1

[5, Corollary 2]* Let $C_{n_{k}}$, $n_{k}\geq 2$, $1\leq k\leq r$ be cycles. Then*

[TABLE]

Moreover, if $n_{k}$ is even for every $1\leq k\leq r$, then

[TABLE]

Problem 1.1

[5, Remark 2]* Given an Abelian group $G$ and an inverse closed minimal generating set $S\subseteq G\setminus 1$ of $G$, is it true that*

[TABLE]

where $S^{*}\subseteq S$ is a minimal generating set of $G$.

In this paper, we improve the upper bound of ${\rm src}(C_{n_{1}}\square\cdots\square C_{n_{r}})$ in Theorem 1.1. Our main result is listed below.

Theorem 1.2

Let $C_{n_{k}}$, $n_{k}\geq 2$, $1\leq k\leq r$ be cycles. Then

[TABLE]

where $\mu$ is the number of even numbers among $n_{1},\ldots,n_{r}$.

Note that an $r$-dimensional undirected toroidal mesh is a Cayley graph. As a result, Theorem 1.2 gives a negative answer to Problem 1.1.

2 Preliminary results

In this section, we will introduce some useful results for the strong rainbow connection numbers of graphs.

Lemma 2.1

[3, Proposition 2.1]* For each integer $n\geq 4$, ${\rm rc}(C_{n})={\rm src}(C_{n})=\left\lceil\frac{n}{2}\right\rceil$.*

We make the following simple observation, which we will use repeatedly.

Observation 2.2

Let $\Gamma$ and $\Lambda$ be two connected graphs. Then

[TABLE]

Lemma 2.3

For each integer $n\geq 3$, ${\rm src}(C_{n}\square C_{2})=\left\lceil\frac{n+1}{2}\right\rceil$.

*Proof. *Write $(0,1,\ldots,n-1)$ for $C_{n}$ and $(0,1)$ for $C_{2}$. Since the diameter of $C_{n}\square C_{2}$ is $\left\lceil\frac{n+1}{2}\right\rceil$, it suffices to show that ${\rm src}(C_{n}\square C_{2})\leq\left\lceil\frac{n+1}{2}\right\rceil$. We only need to construct a strong rainbow $\left\lceil\frac{n+1}{2}\right\rceil$-coloring. Now we divide our discussion into two cases.

Case 1   $n=2k$.

Define an edge-coloring $f_{1}$ of the graph $C_{2k}\square C_{2}$ by

[TABLE]

For illustration, we give a strong rainbow $5$-coloring of $C_{8}\square C_{2}$ in Figure 1.

Note that any path from $u$ to $v$ in Table 1 is a strong rainbow path under the coloring $f_{1}$. It follows that $f_{1}$ is a strong rainbow $(k+1)$-coloring.

Case 2   $n=2k+1$.

Define an edge-coloring $f_{2}$ of the graph $C_{2k+1}\square C_{2}$ by

[TABLE]

For illustration, we give a strong rainbow $4$-coloring of $C_{7}\square C_{2}$ in Figure 2.

Note that any path from $u$ to $v$ in Table 2 is a strong rainbow path under the coloring $f_{2}$. It follows that $f_{2}$ is a strong rainbow $(k+1)$-coloring. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

In the graph $\Gamma\square\Lambda$, we write $\Gamma y$ for $\Gamma\square\{y\}$, where $y\in V(\Lambda)$. The union of graphs $\Gamma$ and $\Lambda$ is the graph $\Gamma\cup\Lambda$ with vertex set $V(\Gamma)\cup V(\Lambda)$ and edge set $E(\Gamma)\cup E(\Lambda)$.

Proposition 2.4

Let $\Gamma$ be a connected graph. Then

[TABLE]

*Proof. *Write $(x_{1},x_{2},\ldots,x_{n})$ for $C_{n}$. Meanwhile, we use $E[\Gamma x_{i},\Gamma x_{j}]$ to denote the edge set between $\Gamma x_{i}$ and $\Gamma x_{j}$.

By Observation 2.2, we have

[TABLE]

Thus, (2) holds for $n=3$. Now, we suppose that $n\geq 4$.

Let $L_{1}$ and $L_{2}$ be induced subgraphs of $\Gamma\square C_{n}$ whose vertex sets are $V(\Gamma x_{1})\cup V(\Gamma x_{n})$ and $V(\Gamma x_{\left\lfloor\frac{n}{2}\right\rfloor})\cup V(\Gamma x_{\left\lfloor\frac{n}{2}\right\rfloor+1})$ respectively. Then each $L_{i}$ is isomorphic to $\Gamma\square C_{2}$. Let $S_{0}=\{1,\ldots,\left\lceil\frac{n-2}{2}\right\rceil\}$. Suppose that $f_{1}:E(\Gamma)\rightarrow S_{1}$ is a strong rainbow ${\rm src}(\Gamma)$-coloring of $\Gamma$, and $f_{2,i}:E(L_{i})\rightarrow S_{2}$ is a strong rainbow ${\rm src}(\Gamma\square C_{2})$-coloring of $L_{i}$ for $1\leq i\leq 2$, where $S_{0}\cap S_{2}=\emptyset$. By Observation 2.2, we may assume that $S_{1}\subseteq S_{2}$. Define an edge-coloring $f_{3}:E(\Gamma\square C_{n})\rightarrow S_{0}\cup S_{2}$ by

[TABLE]

where $I=\{2,3,\ldots,\left\lfloor\frac{n}{2}\right\rfloor-1,\left\lfloor\frac{n}{2}\right\rfloor+2,\ldots,\left\lfloor\frac{n}{2}\right\rfloor+3,\ldots,n-1\}$. For illustration of $f_{3}$, see Figure 3.

Pick any two distinct vertices $u$ and $v$ of $\Gamma\square C_{n}$. Write $u=y_{1}x_{i}$ and $v=y_{2}x_{j}$. Without loss of generality, we may assume that $i\leq j$. We only need to show that there exists a strong rainbow path from $u$ to $v$ under $f_{3}$. If $i=j$, the desired result is obvious. Assume that $i\neq j$. We divide our discussion into three cases.

Case 1   $1\leq j-i\leq\left\lfloor\frac{n}{2}\right\rfloor$, and $2\leq j\leq\left\lfloor\frac{n}{2}\right\rfloor$ or $\left\lfloor\frac{n}{2}\right\rfloor+1\leq i\leq n-1$.

Pick a strong rainbow path $P_{1}$ from $y_{1}x_{j}$ to $v$ in $\Gamma x_{j}$. Then

[TABLE]

is a desired strong rainbow path.

Case 2   $1\leq j-i\leq\left\lfloor\frac{n}{2}\right\rfloor$, $j\geq\left\lfloor\frac{n}{2}\right\rfloor+1$ and $i\leq\left\lfloor\frac{n}{2}\right\rfloor$.

Pick a strong rainbow path $P_{2}$ from $y_{1}x_{\left\lfloor\frac{n}{2}\right\rfloor}$ to $y_{2}x_{\left\lfloor\frac{n}{2}\right\rfloor+1}$ in $L_{2}$. Then

[TABLE]

is a desired strong rainbow path.

Case 3   $j-i\geq\left\lfloor\frac{n}{2}\right\rfloor+1$.

Pick a strong rainbow path $P_{3}$ from $y_{1}x_{1}$ to $y_{2}x_{n}$ in $L_{1}$. Then

[TABLE]

is a desired strong rainbow path.

As mentioned above, we obtain the desired result. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

3 Proof of Theorem 1.2

Proposition 3.1

Let $C_{n_{k}}$, $n_{k}\geq 2$, $1\leq k\leq r$ be cycles. Then

[TABLE]

*Proof. *Without loss of generality, we may assume that $n_{1}\geq n_{2}\geq\cdots\geq n_{r}$. We distinguish two cases.

Case 1   $n_{r}\geq 3$.

We prove this proposition by induction on $r$. If $r=1$, (3) is derived from Lemma 2.1. Suppose $r=2$. If $n_{2}$ is even, then by Lemma 2.1 and Observation 2.2, (3) holds. If $n_{2}$ is odd, then by Proposition 2.4 and Lemma 2.3, we have

[TABLE]

Now, Suppose $r\geq 3$.

If each $n_{i}$ is odd, then

[TABLE]

If $n_{i}$ is even for some $i$, then

[TABLE]

Case 2   $n_{r}=2$.

Suppose $s$ is the minimum positive integer such that $n_{s}=2$.

Case 2.1   $s=1$. By Observation 2.2, (3) is obtained.

Case 2.2   $s\geq 2$. In this case, we have

[TABLE]

Combining Case $1$ and Case $2$, we obtain the desired result. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

*Proof of Theorem 1.2: * If $r=1$, (1) is obvious. Now, suppose $r\geq 2$. We divide our discussion into three cases.

Case 1   $r=2$.

If $n_{1}=n_{2}=2$, Observation 2.2 implies that (1) holds. Now suppose $n_{j}\geq 3$, for some $j$. Without loss of generality, we may assume that $n_{1}\geq 3$. By Proposition 2.4 and Lemma 2.3, we have

[TABLE]

If $\mu=1$ and $n_{1}$ is even, then

[TABLE]

Otherwise, (4) implies (1).

Case 2   $r=3$.

If $\mu=0$ or $\mu=3$, (1) holds by Proposition 3.1. Now suppose $\mu=1$ or $\mu=2$. Without loss of generality, we assume that $n_{1}$ is even and $n_{3}$ is odd. By Observation 2.2, Lemma 2.1 and Case 1, we have

[TABLE]

Case 3   $r\geq 4$.

If $\mu=0$ or $\mu=r$, Proposition 3.1 implies (1). In the following, we assume that $n_{1},\ldots,n_{\mu}$ are even and $n_{\mu+1},\ldots,n_{r}$ are odd.

Case 3.1   $1\leq\mu\leq\left\lfloor\frac{r}{2}\right\rfloor-1$.

[TABLE]

Case 3.2   $\mu=\left\lfloor\frac{r}{2}\right\rfloor$.

[TABLE]

Case 3.3   $\left\lfloor\frac{r}{2}\right\rfloor+1\leq\mu\leq r-1$.

[TABLE]

As mentioned above, we obtain the desired result. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Acknowledgement

M. Xu’s research is supported by the National Natural Science Foundation of China (11571044, 61373021). K. Wang’s research is supported by the National Natural Science Foundation of China (11671043, 11371204).