Rainbow Connection Numbers of WK-Recursive Networks and WK-Recursive Pyramids

Abstract

An edge coloring of a graph G results in G being rainbow connected when every pair of vertices is linked by a rainbow path. Such a path is defined as one where each edge possesses a distinct color. A rainbow coloring refers to an edge coloring that guarantees the rainbow connectedness of G. The rainbow connection number of G represents the smallest quantity of colors required to achieve rainbow connectedness under a rainbow coloring scheme. Wang and Hsu (ICICM 2019: 75–79) provided upper bounds on the size of the rainbow connection numbers in WK-recursive networks ${\mathrm{WK}}_{d,t}$ and WK-recursive pyramids ${\mathrm{WKP}}_{d,n}$. In this paper, we revise their results and determine the exact values of the rainbow connection numbers of ${\mathrm{WK}}_{d,2}$ for $d=3$ and 4. The rainbow connection numbers of ${\mathrm{WK}}_{d,2}$ are bounded between 4 and $\lfloor \frac{d}{2}\rfloor +2$ for $d>4$. In addition to our previous findings, we further investigate and determine upper bounds for the size of the rainbow connection numbers of ${\mathrm{WKP}}_{d,n}$. This involves analyzing various aspects of the graph structure and exploring potential limitations on the rainbow connection numbers. By establishing these upper bounds, we gain deeper insights into the potential range and constraints of the rainbow connection numbers within the given context.

1. Introduction

When representing interconnection networks, it is customary to employ graphs wherein the vertices represent the processing nodes within the system and the edges signify the communication links between these nodes. The process of assigning colors to the edges of a graph G is known as edge coloring, which is essentially a function mapping the edge set $E\left(G\right)$ to the set of natural numbers. This coloring scheme enables us to distinguish between different communication paths within the network based on the colors assigned to their corresponding edges. A path$P=v_1\to v_2\to \cdots \to v_i$ in G is defined as either a single vertex or a sequence of distinct vertices, where each consecutive pair of vertices $v_1\to v_2$, $v_2\to v_3,\dots ,v_{i-1}\to v_i$ forms an edge in G. In a graph G, where u and v are distinct vertices belonging to the vertex set $V\left(G\right)$, a path linking u and v is defined as a $u-v$path. In an edge-colored graph G, where neighboring edges might share the same color, a $u-v$path is termed a rainbow $u-v$ path if none of its edges share the same color. Consequently, an edge-colored graph G is rainbow connected if every pair of vertices u and v is connected by a rainbow $u-v$ path. An edge coloring c that ensures the rainbow connectedness of G is referred to as a rainbow coloring. If k colors are utilized in this coloring scheme, then c is recognized as a rainbow k-coloring. The rainbow connection number of a connected graph G, denoted as rc(G), represents the smallest number of colors required to render G rainbow connected.

The concept of rainbow coloring in graphs was first introduced by Chartrand et al. [1] and has garnered significant attention due to its practical applications, particularly around ensuring the secure transfer of classified information among various agencies [2]. This method is particularly crucial in scenarios where intermediary agencies are involved, as it facilitates secure communication by assigning unique passwords between the agencies. When pairs of agencies establish one or more secure paths, each with its distinct set of passwords, this establishes a secure network and reveals the rainbow connection inherent within the system. This revelation acts as a formidable deterrent against potential intruders, enhancing the overall security posture of the communication network.

Extensive discussions surrounding the problem and its myriad applications have been meticulously explored from a combinatorial perspective, as evidenced by the publication of over 150 papers dedicated to this subject matter. Furthermore, ongoing research endeavors continue to yield significant advancements, particularly in the exploration of various variants of rainbow coloring across diverse graph families. For a comprehensive overview of these developments, interested readers can refer to authoritative sources such as surveys [3,4] and a dedicated book [5], which provide insightful insights and analyses into the evolving landscape of rainbow coloring within graph theory.

Extensive research has been conducted on the rainbow connection number and rainbow coloring, exploring these concepts from both algorithmic and graph-theoretic perspectives. Scholars have delved into various aspects of these topics, investigating their theoretical foundations as well as their practical implications. Notably, Chakraborty et al. demonstrated that determining the rainbow connection number of a general graph is NP-hard [6]. In fact, even deciding whether $\mathrm{rc}\left(G\right)=2$ is already known to be an NP-complete problem. Similarly, determining whether a given edge-colored graph is rainbow connected is also NP-complete [6]. An easy observation is that $\mathrm{rc}\left(G\right)=1$ if and only if G is a complete graph. In [7], Caro et al. provided valuable insights into the conditions under which the rainbow connection number of a graph G equals 2. Chartrand et al. investigated the rainbow connection numbers of a diverse range of graph structures, including complete graphs, trees, cages, wheel graphs, bipartite graphs, and multipartite graphs [1,8]. Over the last decade, several researchers have studied the bounds of the rainbow connection numbers on 3-connected graphs [9], Cayley graphs [10], the comb product of graphs [11], the power graph of a finite group [12,13], and several graph operations such as the Cartesian product, lexicographic product, strong product, and union [14,15].

This paper directs its focus towards elucidating the methodologies involved in constructing rainbow colorings for both a given WK-recursive network and a WK-recursive pyramid. WK-recursive networks, initially introduced in [16], represent a class of interconnection networks characterized by their recursive scalability. These networks are constructed hierarchically through the grouping of basic modules, resulting in structures that offer remarkable regularity and scalability. This inherent modular design of WK-recursive networks aligns well with the requirements of distributed systems, as highlighted in [16,17], comprising a multitude of computing elements. Building upon the foundation laid down by WK-recursive networks, the WK-recursive pyramid emerges as an interconnection network tailored for massively parallel computers, as outlined in [18]. Utilizing the innate hierarchy at each tier, WK-recursive pyramids demonstrate adeptness in addressing a spectrum of issues in graph theory, digital geometry, image processing, and machine vision [18]. Additionally, the fault-tolerant characteristics of WK-recursive pyramids position them as promising networks for dependable computing [18]. In addition to their rainbow connectivity, WK-recursive pyramids exhibit several other advantageous properties. Notably, they boast small average distances and diameters along with substantial connectivity, scalability, and expandability. These attributes render WK-recursive pyramids highly suitable for both medium and large networks, presenting an optimal choice for scenarios in which efficiency and robustness are paramount considerations. Moreover, when compared to conventional mesh and traditional pyramid interconnection topologies, WK-recursive pyramids emerge as the superior alternative, offering enhanced performance and flexibility for a diverse range of applications, as highlighted in [18]. Wang and Hsu [19] provided upper bounds on the size of the rainbow connection numbers in WK-recursive networks and WK-recursive pyramids. In this paper, we revise and improve their results.

The remainder of this article is organized as follows. Section 2 delves into the rainbow connection numbers of ${\mathrm{WK}}_{d,t}$. Building upon the findings regarding WK-recursive networks, Section 3 extends the investigation further to provide and determine upper bounds for the size of the rainbow connection numbers of ${\mathrm{WKP}}_{d,n}$. In the final section of this paper, we provide concluding remarks that summarize the key findings and contributions discussed throughout the study.

2. Results on WK-Recursive Networks

For $d\in \mathbb{N}$, the set $\{0,1,\cdots ,d-1\}$ is shortly denoted by $\left[d\right]$ and the complement of a subset S of $\left[d\right]$ is denoted by ${\left[d\right]}_S$.

Definition 1

([16]). ${\mathit{WK}}_{d,t}$ is a WK-recursive network with amplitude d and expansion level t. Vertex $v=a_{t-1}{a}_{t-2}\cdots a_0\in V\left({WK}_{d,t}\right)$ is labeled as a t-digit radix d number. An edge $⟨v,a_{t-1}{a}_{t-2}\cdots b⟩,b\in {\left[d\right]}_{a_0}$ is called a substituting edge. A vertex $a_{t-1}{a}_{t-2}\dots a_j{\left(x\right)}^j$ is called a j-frontier. An edge $⟨v,a_{t-1}{a}_{t-2}\dots a_j{\left(x\right)}^j⟩$ is said to be a j-flipping edge. An open edge is an edge incident on ${\left(a_{t-1}\right)}^t$, and ${\left(a_{t-1}\right)}^t$ is called a corner vertex.

The open edge is reserved for further expansion; hence, its other end vertex is unspecified. For a vertex $v\in V\left({\mathrm{WK}}_{d,t}\right)$, let ${\sigma }_i\left(v\right)$ and ${\rho }_i\left(v\right)$ denote the leftmost i digits and the i-th rightmost digit of the address of v, respectively (that is, ${\sigma }_i\left(v\right)=a_{t-1}{a}_{t-2}\dots a_{t-i}$ and ${\rho }_i\left(v\right)=a_{i-1}$); in particular, ${\sigma }_0\left(v\right)=\varnothing$. For any j-flipping edge $⟨v,v^{\prime }⟩$, ${\rho }_{j+1}\left(v\right)={\rho }_1\left(v^{\prime }\right)$. A vertex can be denoted using a sequence of symbols drawn from the set $\left[d\right]\cup \{\ast \}$, where ∗ is a “don’t care” symbol. The labeling represented by $a_{t-1}{a}_{t-2}\dots a_{j+1}\ast a_{j-1}\dots a_1{a}_0$ depicts a set of vertices where the addresses differ solely in the digit $a_j$.

For vertices u and v in G, the distance between u and v, denoted ${\mathrm{dist}}_G(u,v)$, is the length of the shortest $u-v$ path in G. The diameter $\mathrm{diam}\left(G\right)$ of a graph G is defined as the maximum distance over all pairs of vertices in G. For a connected graph G, $\mathrm{rc}\left(G\right)\ge \mathrm{diam}\left(G\right)$, where $\mathrm{diam}\left(G\right)$ denotes the diameter of G. The inequality is true because any rainbow path between two vertices of distance $\mathrm{diam}\left(G\right)$ must use at least $\mathrm{diam}\left(G\right)$ different colors. ${\mathrm{WK}}_{d,t}$ is a class of recursively scalable networks of $\mathrm{diam}\left({\mathrm{WK}}_{d,t}\right)=2^t-1$ [20]. We define $a_{t-1}{a}_{t-2}\dots a_i·{\mathrm{WK}}_{d,i}$ as a subgraph of ${\mathrm{WK}}_{d,t}$ induced by the set of vertices $a_{t-1}{a}_{t-2}\dots a_i{(\ast )}^i$, that is, $a_{t-1}{a}_{t-2}\dots a_i·{\mathrm{WK}}_{d,i}$ is an embedded ${\mathrm{WK}}_{d,i}$ with the identifier $a_{t-1}{a}_{t-2}\dots a_i$. Figure 1 depicts ${\mathrm{WK}}_{4,2}$ and ${\mathrm{WK}}_{4,3}$. In Figure 1a, the edge $⟨01,02⟩$ is identified as a substituting edge, whereas the edge $⟨01,10⟩$ serves as the 1-flipping edge, connecting the 1-frontiers 01 and 10. Open edges are incident with the corner vertices $00,11,22$, and 33. The values of ${\sigma }_1\left(13\right),{\sigma }_2\left(13\right),{\rho }_1\left(13\right)$, and ${\rho }_2\left(13\right)$ are $1,13,3$, and 1, respectively. In Figure 1b, $0·{\mathrm{WK}}_{4,2}$ is a subgraph of ${\mathrm{WK}}_{4,3}$ induced by the set of vertices $0{(\ast )}^2$. ${\mathrm{WK}}_{4,3}$ has the 2-flipping edges $⟨011,100⟩,⟨122,211⟩,⟨233,322⟩,⟨300,033⟩,⟨022,200⟩$, and $⟨133,311⟩$.

Figure 1. Rainbow colorings of (a) ${\mathrm{WK}}_{4,2}$ and (b) ${\mathrm{WK}}_{4,3}$.

Chartrand et al. provided the rainbow connection numbers of complete graphs, trees, and cycles as the following results [1].

Proposition 1

([1]).

 (a) 

$rc\left(G\right)=\left|V\right(G\left)\right|-1$ if and only if G is a tree.

 (b) 

$rc\left(G\right)=1$ if and only if G is a complete graph.

 (c) 

$rc\left(G\right)=\lceil \frac{n}{2}\rceil$ if G is a cycle of order $n\ge 4$.

Because ${\mathrm{WK}}_{2,t}$ is in fact a path of length $2^t-1$, the next result follows directly from Proposition 1(a).

Corollary 1.

For $t\ge 1$, $rc\left({WK}_{2,t}\right)=2^t-1$.

Because the diameter of a graph is a trivial lower bound, $\mathrm{rc}\left({\mathrm{WK}}_{d,t}\right)\ge 2^t-1$. In standard practice, the color number assigned to an edge is typically denoted as ${\left(i\right)}_d$, where i represents the string of digits and d denotes the base in which it is expressed. This notation allows for clear and concise representation of the assigned color, facilitating ease of interpretation and communication within the context of edge coloring in graphs.

Theorem 1.

For $t\ge 1$, $2^t-1\le rc\left({WK}_{d,t}\right)\le d^{t-1}$, where $d\ge 3$.

Proof. 

We provide an edge coloring $c_w:E\left({\mathrm{WK}}_{d,t}\right)\to \{0,1,\dots ,d^{t-1}-1\}$ of ${\mathrm{WK}}_{d,t}$. For a vertex $v=a_{t-1}{a}_{t-2}\dots a_1{a}_0$, let all substituting edges incident on v be assigned the color ${\left({\sigma }_{t-1}\left(v\right)\right)}_d$. Then, $d^{t-1}$ are needed to color the substituting edges. Taking Figure 1b as an example, every substituting edge incident on the vertex 310 is assigned the color ${\sigma }_2\left(310\right)=31$. The color assigned to a j-flipping edge $e=⟨v,a_{t-1}{a}_{t-2}\dots a_{j+1}{a}_{j-1}{\left(a_j\right)}^j⟩$ is defined by

$c_w\left(e\right)=\left\{\begin{array}{cc}{\left({\sigma }_{t-j-1}\left(v\right)0{\left(0\right)}^{j-1}\right)}_d\hfill & \mathrm{if}{a}_j\ne 0\mathrm{and}{a}_{j-1}\ne 0,\hfill \\ {\left({\sigma }_{t-j-1}\left(v\right)1{\left(0\right)}^{j-1}\right)}_d\hfill & \mathrm{if}(a_j=0\mathrm{and}{a}_{j-1}\ne 1)\mathrm{or}(a_{j-1}=0\mathrm{and}{a}_j\ne 1),\hfill \\ {\left({\sigma }_{t-j-1}\left(v\right)2{\left(0\right)}^{j-1}\right)}_d\hfill & \mathrm{if}(a_j=0\mathrm{and}{a}_{j-1}=1)\mathrm{or}(a_{j-1}=0\mathrm{and}{a}_j=1).\hfill \end{array}\right.$

Again using Figure 1b as an example, the 1-flipping edge $e=⟨001,010⟩$ is assigned the color ${\sigma }_1\left(001\right)2=02$ and the 2-flipping edge $e=⟨011,100⟩$ is assigned the color ${\sigma }_0\left(011\right)2{\left(0\right)}^1=20$. The colors assigned to the edges of the subnetwork $a_{t-1}{a}_{t-2}\dots a_{j+1}·{\mathrm{WK}}_{d,j+1}$ range from ${(a_{t-1}{a}_{t-2}\dots a_{j+1}{\left(0\right)}^j)}_d$ to ${(a_{t-1}{a}_{t-2}\dots a_{j+1}{(d-1)}^j)}_d$. We employ an inductive approach in order to show that $c_w$ constitutes a rainbow coloring of ${\mathrm{WK}}_{d,t}$, specifically working on induction with respect to t. When $t=1$, ${\mathrm{WK}}_{d,1}$ takes the form of a complete graph with d vertices. According to Proposition 1(b), $\mathrm{rc}\left({\mathrm{WK}}_{d,1}\right)=1$. We assume that this holds true for $t=k>1$, and now proceed to examine the case when $t=k+1$. Recall that $0·{\mathrm{WK}}_{d,k},1·{\mathrm{WK}}_{d,k},\dots ,(d-1)·{\mathrm{WK}}_{d,k}$ are d copies of ${\mathrm{WK}}_{d,k}$ embedded in ${\mathrm{WK}}_{d,k+1}$. Fro the inductive hypothesis, each $a_k·{\mathrm{WK}}_{d,k}$ is colored by using $d^{k-1}$ colors. Let $u=u_k{u}_{k-1}\dots u_0$ and $v=v_k{v}_{k-1}\dots v_0$ be two nonadjacent vertices of ${\mathrm{WK}}_{d,k+1}$. If $u_k=v_k$, then u and v are in the same subnetwork $u_k·{\mathrm{WK}}_{d,k}$; thus, according to the inductive hypothesis, the assertion holds. Conversely, u and v reside in distinct subnetworks, with u belonging to subnetwork $u_k·{\mathrm{WK}}_{d,k}$ and v to subnetwork $v_k·{\mathrm{WK}}_{d,k}$. From the inductive hypothesis, we have a rainbow $u-u_k{\left(v_k\right)}^k$ path $P_1$ with edge colors ranging from ${\left(u_k{\left(0\right)}^{k-1}\right)}_d$ to ${\left(u_k{(d-1)}^{k-1}\right)}_d$, and a rainbow $v-v_k{\left(u_k\right)}^k$ path $P_2$ with edge colors ranging from ${\left(v_k{\left(0\right)}^{k-1}\right)}_d$ to ${\left(v_k{(d-1)}^{k-1}\right)}_d$. Our objective at this point is to demonstrate that the color number assigned to the k-flipping edge $e=⟨u_k{\left(v_k\right)}^k,v_k{\left(u_k\right)}^k⟩$ is not in $S=\{{\left(u_k{\left(0\right)}^{k-1}\right)}_d,{\left(u_k{\left(0\right)}^{k-2}1\right)}_d,\dots ,{\left(u_k{(d-1)}^{k-1}\right)}_d,{\left(v_k{\left(0\right)}^{k-1}\right)}_d,{\left(v_k{\left(0\right)}^{k-2}1\right)}_d,\dots ,{\left(v_k{(d-1)}^{k-1}\right)}_d\}$. If $u_k\ne 0\mathrm{and}{u}_{k-1}\ne 0$, then $c_w\left(e\right)={\left(0{\left(0\right)}^{k-1}\right)}_d$. Clearly, $c_w\left(e\right)\notin S$. For the case $(u_k=0\mathrm{and}{u}_{k-1}\ne 1)\mathrm{or}(u_{k-1}=0\mathrm{and}{u}_k\ne 1)$, we obtain $c_w\left(e\right)={\left(2{\left(0\right)}^{k-1}\right)}_d\notin S$. When $(u_k=0\mathrm{and}{u}_{k-1}=1)\mathrm{or}(u_{k-1}=0\mathrm{and}{u}_k=1)$, we have $c_w\left(e\right)={\left(1{\left(0\right)}^{k-1}\right)}_d\notin S$. Therefore, the concatenation of the path $P_1$, k-flipping edge e, and path $P_2$ constructs a rainbow $u-v$ path. □

The exact value of $\mathrm{rc}\left({\mathrm{WK}}_{3,2}\right)$ is deduced directly from Theorem 1.

Corollary 2.

$rc\left({WK}_{3,2}\right)=3$.

Furthermore, we aim to narrow down the difference between the lower and upper bounds of $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)$ for $d\ge 4$.

Theorem 2.

For $d\ge 4$, $4\le rc\left({WK}_{d,2}\right)\le \lceil \frac{d}{2}\rceil +2$.

Proof. 

Because diam$\left(\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)\right)=3$, we have $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)\ge 3$. To show the lower bound of $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)$, we can assume, to the contrary, that $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)=3$ for $d\ge 4$; that is, any two vertices u and v are connected by a rainbow $u-v$ path of length $\le 3$. Let u be the corner vertex 00 and let $v=1x,x\in {\left[d\right]}_0$. Because $00\to 01\to 10\to 1x$ is the unique rainbow $u-v$ path, the edges $⟨00,01⟩,⟨01,10⟩$ and $⟨10,1x⟩$ are assigned with three distinct colors (0, 1, and 2). Without loss of generality, let $⟨10,1x⟩$ be assigned the color 1, that is, every substituting edge incident with 10 is colored 1. Similarly, for each $y\in {\left[d\right]}_1$, every substituting edge incident with $1y$ is colored 1 to reveal the fact that $yy\to y1\to 1y\to 1z$ is the unique rainbow $22-1z$ path, where $z\in {\left[d\right]}_y$. This implies that every edge of $1·$ ${\mathrm{WK}}_{d,1}$ is colored 1. For the same reason, all edges of a subnetwork $a·$${\mathrm{WK}}_{d,1}$, $a\in \left[d\right]$ must be assigned the same color, and this must be distinct from the color assigned to the edges of $b·$${\mathrm{WK}}_{d,1}$, where $b\in {\left[d\right]}_a$. Thus, $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)\ge d$, which is a contradiction. We now construct an edge coloring $c:E\left({\mathrm{WK}}_{d,2}\right)\to \{0,1,\dots ,\lceil \frac{d}{2}\rceil +1\}$ to prove $\mathrm{rc}\left({\mathrm{WK}}_{d,2}\right)\le \lceil \frac{d}{2}\rceil +2$. A d-cycle $C_a:a_0\to a_1\to \cdots \to a_{d-1}\to a_0$ is called an outer cycle of $a·{\mathrm{WK}}_{d,1}$. For every $C_a$, we refer the rainbow coloring of Proposition 1(c) and let the edges of $C_a$ be assigned the colors range from 0 to $\lceil \frac{d}{2}\rceil -1$. Let each substituting edge not in an outer cycle be assigned the color $\lceil \frac{d}{2}\rceil$ and every flipping edge not in an outer cycle be assigned the color $\lceil \frac{d}{2}\rceil +1$. We claim that the edge coloring c is a rainbow $(\lceil \frac{d}{2}\rceil +2)$-coloring of ${\mathrm{WK}}_{d,2}$. Let $u=u_1{u}_0$ and $v=v_1{v}_0$ be two nonadjacent vertices of ${\mathrm{WK}}_{d,2}$ and $C_{u_1}$, and let $C_{v_1}$ be the outer cycle containing u and v, respectively. Recall that vertices $u_1{v}_1$ and $v_1{u}_1$ are both frontiers and that the flipping edge $⟨u_1{v}_1,v_1{u}_1⟩$ is assigned the color $\lceil \frac{d}{2}\rceil +1$ in c. We now show that u and v are connected by a rainbow $u-v$ path. Because u and v are not adjacent, $u_1\ne v_1$ and $u,v$ cannot both be frontiers. If u is a frontier, then $u\to v_1{v}_2\to v$ is a rainbow path due to $c\left(⟨u,v_1{v}_2⟩\right)=\lceil \frac{d}{2}\rceil +1$ and $c\left(⟨v_1{v}_2,v⟩\right)=\lceil \frac{d}{2}\rceil$. For the same reason, $v\to u_1{u}_2\to u$ is a rainbow path when v is a frontier. For the case in which neither u nor v is a frontier, let P be a path from the vertex $v_1{u}_1$ to v such that P is a subpath of the outer cycle of $v·{\mathrm{WK}}_{d,1}$. Because P is a rainbow path and the edge of P is assigned the color $\le \lceil \frac{d}{2}\rceil$, the concatenation of the path $u\to u_1{v}_1\to v_1{u}_1$ and P is a rainbow $u-v$ path. □

The exact value of $\mathrm{rc}\left({\mathrm{WK}}_{4,2}\right)$ follows directly.

Corollary 3.

$rc\left({WK}_{4,2}\right)=4$.

3. Results on WK-Recursive Pyramids

In this section, we present our methodology for edge coloring of WK-recursive pyramids, aiming to determine an upper bound for their rainbow connection numbers. Our approach involves developing a systematic edge coloring scheme tailored specifically for WK-recursive pyramids. Before delving into the coloring process, we provide a comprehensive definition of WK-recursive pyramids, elucidating their structural characteristics and recursive construction principles.

Definition 2.

An n-dimensional WK-recursive pyramid [18], denoted ${\mathit{WKP}}_{d,n}$, consists of a set of vertices $\left\{\left(0\right)\right\}\cup \left\{(l;a_{l-1}{a}_{l-2}\dots a_0)\right|1\le l\le n,a_i\in \left[d\right],0\le i\le l-1\}$. The subgraph induced by the vertex set $\left\{(l;a_{l-1}{a}_{l-2}\dots a_0)\right|a_i\in \left[d\right],0\le i\le l-1\}$, denoted $L_l$, is connected as a ${\mathit{WK}}_{d,l}$ and called layer l. The apex $\left(0\right)$ is $L_0$. A vertex v with the addressing scheme $(l;a_{l-1}{a}_{l-2}\dots a_0)$ is a vertex at layer l. The part $a_{l-1}{a}_{l-2}\dots a_0$ of the address determines the address of v within $L_l$. The children of v are the d vertices with the address scheme $(l+1;a_{l-1}{a}_{l-2}\dots a_{0}i),i\in \left[d\right]$. The parent vertex of v, denoted by $p\left(v\right)$, is the vertex $(l-1;a_{l-1}{a}_{l-2}\dots a_1)$.

An edge incident to a vertex and its parent vertex is called a vertical edge, while every edge within a layer is called a horizontal edge. ${\mathrm{WKP}}_{d,1}$ is a complete graph of order $d+1$. Throughout the remainder of the paper, we focus our analysis on cases where n is greater than or equal to 2. A vertex $u\ne v$ is an ancestor of v if u lies along the shortest path from v to the apex of the pyramid. ${\mathrm{WKP}}_{d,n}$ has a small diameter $2n-1$, while the $L_i$ of ${\mathrm{WKP}}_{d,n}$ for each large i has a large diameter $2^i-1$. The spirit of our algorithm is based on the following proposition.

Proposition 2.

Let $u,v\in V\left(L_i\right)$, $i\le n$; if ${\mathit{dist}}_{L_i}(u,v)>rc\left({WKP}_{d,n}\right)$, then in every rainbow $u-v$ path there are an even number of vertical edges between layers $i-1$ and i.

We propose an edge coloring $c_p$ on ${\mathrm{WKP}}_{d,n}$ to establish an upper bound of $\mathrm{rc}\left({\mathrm{WKP}}_{d,n}\right)$. Let $c_p\left(e\right)$ denote the color number assigned to the edge e and $c_p\left(E\right)={\bigcup }_{e\in E}{c}_p\left(e\right)$ for an edge set E. For a dedicated layer number $\tau$, the layers $0,1,\dots ,\tau$ are called small layers, while the layers $\tau +1,\tau +2,\dots ,n$ are said to be large layers. For every two vertices u and v of a small layer L, we need to ensure that there is a rainbow $u-v$ path containing only horizontal edges. If L is a large layer, then, from Proposition 2, we want to establish a rainbow $u-v$ path that contains even number of vertical edges. The edge coloring of a small layer $L_i$ follows from the rainbow coloring $c_w$ defined in Section 2, while $c_p\left(E\left(L_i\right)\right)=\{0,1,\dots ,d^{i-1}\}$ for each small layer $L_i$. Let the vertical edges between small layers $i-1$ and i be assigned a color $(n-\tau )\ast d^{n-\lambda }+\tau -i$, and let each large layer $L_i$ be partitioned into $d^{n-\lambda }$ subnetworks $1\le \lambda \le n-1$ of expansion level $\lambda -(n-i)$. Every subnetwork with an address scheme $(i;a_{i-1}{a}_{i-2}\dots a_{\lambda -(n-i)}·{\mathrm{WK}}_{d,\lambda -(n-i)})$ in $L_i$ is regarded as a basic colored module and has a rainbow $d^{\lambda -(n-i)-1}$-coloring $c_w$. All horizontal edges between two basic colored modules are not in any rainbow path, and can be left empty or simply assigned a color 0. We let all vertical edges incident on the vertices in $(i;a_{i-1}{a}_{i-2}\dots a_{\lambda -(n-i)}·{\mathrm{WK}}_{d,\lambda -(n-i)})$ and their parent vertices be assigned identical colors, and assign the vertical edges incident on two distinct subnetworks of a layer distinct colors. Then, $d^{n-\lambda }$ colors are needed for assignment to the vertical edges between large layers, as there are $d^{n-\lambda }$ subnetworks of expansion level $\lambda -(n-i)$ in a large layer. The formal definitions of the edge coloring $c_p$ on the substituting edges, flipping edges, and vertical edges are described as follows.

Definition 3.

Let $e_{h1}=⟨(l;a_{l-1}\dots a_1{a}_0),(l,a_{l-1}\dots a_1{\left[d\right]}_{a_0})⟩$ be a substituting edge,

$e_{h2}=⟨(l;a_{l-1}\dots a_1{a}_0),(l,a_{l-1}\dots a_{j+1}{a}_{j-1}{\left(a_j\right)}^j)⟩$ a j-flipping edge, and

$e_v=⟨u=(l;a_{l-1}\dots a_{n-\lambda }{(\ast )}^{l-(n-\lambda )}),p\left(u\right)⟩$ a vertical edge of ${WKP}_{d,n}$.

$c_p\left(e_{h1}\right)=\left\{\begin{array}{cc}{(a_{l-1}\dots a_1)}_d\hfill & if1\le l<\tau ,\hfill \\ (n-\tau )d^{n-\tau }+{(a_{\tau -1}{a}_{\tau -2}\dots a_1)}_d\hfill & ifl=\tau ,\hfill \\ {(a_{l-(n-\lambda )-1}{a}_{l-(n-\lambda )-2}\dots a_1)}_d\hfill & if\tau <l\le n.\hfill \end{array}\right.$

$c_p\left(e_{h2}\right)=\left\{\begin{array}{cc}{d}^{j-1}+{(a_{l-1}\dots a_j)}_d\hfill & if1\le l<\tau ,\hfill \\ (n-\tau )d^{n-\tau }+d^{j-1}+{(a_{\tau -1}\dots a_j)}_d\hfill & ifl=\tau ,\hfill \\ d^{j-1}+{(a_{l-(n-\lambda )-1}\dots a_j)}_d\hfill & if\tau <l\le n.\hfill \end{array}\right.$

$c_p\left(e_v\right)=\left\{\begin{array}{cc}(n-\tau )d^{n-\tau }+\tau -l\hfill & if1\le l<\tau ,\hfill \\ {(a_{l-1}{a}_{l-2}\dots a_{l-(n-\lambda )})}_d\hfill & if\tau \le l\le n.\hfill \end{array}\right.$

Figure 2 demonstrates the coloring $c_p$ on ${\mathrm{WKP}}_{4,3}$, where $\lambda =2$ and $\tau =2$. ${\mathrm{WKP}}_{4,3}$ has only one large layer $L_3$ under $c_p$. In Figure 2c, $L_3$ is partitioned into $d^{n-\lambda }=4$ subnetworks. Each subnetwork within $L_3$ operates as an independent basic color module, and as such is allocated the identical set of colors. From Theorem 1, the edges of each basic color module are assigned the color numbers in $\{00,01,02,03\}$. The vertical edges $⟨(2;3\ast ),(3;3{(\ast )}^2)⟩$, from Definition 3, are assigned the color ${\left(3\right)}_4$. Here, $L_2$ is the layer $L_{\tau }$. From Definition 3, every horizontal edge of $L_2$ is assigned a color greater than the color numbers used in a large layer (refer to Figure 2b). Then, the horizontal edge of $L_2$ is assigned the color numbers in $\{10,11,12,13\}$. Figure 2a depicts the edge coloring on the small layer $L_1$ which connects to the apex. We now demonstrate that $c_p$ constitutes a rainbow coloring specifically tailored for a WK-recursive pyramid. This involves proving that $c_p$ satisfies the necessary conditions for rainbow connectedness within the structure of the pyramid. Consider the function $l\left(v\right)$, which is defined as the layer number of vertex v.

Figure 2. The rainbow coloring $c_p$ of ${\mathrm{WKP}}_{4,3}$.

Theorem 3.

The edge coloring $c_p$ is a rainbow coloring on ${WKP}_{d,n}$, where $d,n\ge 2$.

Proof. 

Let $s,t\in V\left({\mathrm{WKP}}_{d,n}\right)$. If s and t are on the same subnetwork, then per Theorem 2 there exists a rainbow $s-t$ path. When s and t reside in distinct subnetworks, we explore three cases based on their respective layer numbers $l\left(s\right)$ and $l\left(t\right)$.

• Case 1. $l\left(s\right)=0$.

It is confirmed that vertex s is an ancestor of vertex t, establishing their hierarchical relationship within the network. We define P as the $s-t$ path consisting solely of vertical edges, adhering to the criteria outlined in Definition 3. According to this definition, the edges comprising P are assigned unique color numbers, ensuring that no two edges within P share the same color.

• Case 2. $1\le l\left(s\right)<\tau$.

Consider $a_t$, the ancestor of t located on layer $l\left(s\right)$; according to Definition 3, we can establish the existence of a rainbow $t-a_t$ path, denoted as $P_1$, which exclusively comprises vertical edges. Additionally, because layer $l\left(s\right)$ constitutes a subnetwork, from Theorem 2 we can derive the presence of a rainbow $s-a_t$ path, referred to as $P_2$, consisting solely of layer edges. It is evident that the combination of paths $P_1$ and $P_2$ forms a rainbow $s-t$ path, thereby establishing connectivity between vertices s and t within the network.

• Case 3. $\tau \le l\left(s\right)\le n$.

Consider two vertices $a_s$ and $a_t$ located on layer x, serving as the respective ancestors of vertices s and t. According to Definition 3, we can establish the presence of a rainbow $s-a_s$ path, $P_1$, and a rainbow $t-a_t$ path, $P_2$, both comprised exclusively of vertical edges. Notably, the color numbers associated with $P_1$ and $P_2$ are less than or equal to $(n-\tau )d^{n-\tau }+{(a_{\tau -1}{a}_{\tau -2}\dots a_1)}_d$ and $(n-\tau )d^{n-\tau }+d^{j-1}+{(a_{\tau -1}{a}_{\tau -2}\dots a_j)}_d$, respectively. Because layer $\tau$ corresponds to a ${WK}_{d,\tau }$, as established by Theorem 2, we can deduce the existence of a rainbow $a_s-a_t$ path, denoted as $P_3$, consisting exclusively of layer edges. Remarkably, each layer edge within $P_3$ possesses color numbers that surpass those of any vertical edge in $P_1$ and $P_2$. As a consequence, combining the paths $P_1,P_3$ and $P_2$ yields the formation of a rainbow $s-t$ path. □

The rainbow coloring problem on the network ${\mathrm{WKP}}_{d,n}$ is approached as a minimax optimization challenge. This involves the task of determining an optimal solution, denoted as

$$\underset{\lambda ,\tau }{min}\max (\mathrm{rc}\left({\mathrm{WK}}_{d,\lambda }\right),(n-\tau )d^{n-\lambda }+\mathrm{rc}\left({\mathrm{WK}}_{d,\tau }\right)).$$

Theorem 4.

Let $f=\lfloor \frac{n}{2}\rfloor +ϵ-1$, where $1\le ϵ\le n-1-\lfloor \frac{n}{2}\rfloor$.

$$rc\left({WKP}_{d,n}\right)\le \left\{\begin{array}{cc}{d}^{\lfloor \frac{n}{2}\rfloor }\hfill & if{d}^{\lfloor \frac{n}{2}\rfloor }\ge (\lfloor \frac{n}{2}\rfloor +1)d^{n-(\lfloor \frac{n}{2}\rfloor +1)},\hfill \\ (n-1)d\hfill & if(n-1)d\ge d^{n-2},\hfill \\ min(d^f,f·d^{n-f})\hfill & otherwise.\hfill \end{array}\right.$$

Proof. 

Based on the results on WK-recursive networks, we will determine the value

$$\mathrm{rcn}=\underset{\lambda ,\tau }{min}\max (d^{\lambda -1},(n-\tau )d^{n-\lambda }+d^{\tau -1}).$$

In such situations, we need to decide the values of $\tau$ and $\lambda$. For the large layer $L_{\tau +1}$, if we apply the rainbow coloring $c_w$ on $L_{\tau +1}$ directly, than $d^{\tau }$ colors are needed to color the edges of $L_{\tau +1}$. Actually, $L_{\tau +1}$ is instead partitioned into subnetworks to satisfy the inequality $d^{n-\lambda }+d^{\tau -1}\le d^{\tau }$. Then,

$$\begin{array}{c}{d}^{n-\lambda }\le d^{\tau }-d^{\tau -1}\\ d^{n-\lambda }\le d^{\tau -1}(d-1)\\ d^{n-\lambda -\tau +1}\le d-1\\ n-\lambda -\tau +1<1\\ n<\lambda +\tau .\end{array}$$

(1)

In contrast, the small layer $L_{\tau }$ should not be partitioned due to $d^{n-\lambda }+d^{\tau -2}\ge d^{\tau -1}$. Then,

$$\begin{array}{c}{d}^{n-\lambda }\ge d^{\tau -1}-d^{\tau -2}\\ d^{n-\lambda }\ge d^{\tau -2}(d-1)\\ d^{n-\lambda -\tau +2}\ge d-1\\ n-\lambda -\tau +2>0\\ n+2>\lambda +\tau .\end{array}$$

(2)

From inequalities (1) and (2), we have $\lambda +\tau =n+1$. Then,

$$\mathrm{rcn}=\underset{\lambda }{min}\max (d^{\lambda -1},\lambda d^{n-\lambda }).$$

At this juncture, we are prepared to delve into a discussion regarding the two values $d^{\lambda -1}$ and $\lambda d^{n-\lambda }$. Because $\lambda +\tau =n+1$ and $\lambda >\tau$, $\lambda >\frac{n}{2}$, we let $\lambda =\lfloor \frac{n}{2}\rfloor +ϵ$, where $1\le ϵ\le n-\lfloor \frac{n}{2}\rfloor -1$. We let $x_{ϵ}=d^{\lfloor \frac{n}{2}\rfloor +ϵ-1}$ and $y_{ϵ}=(\lfloor \frac{n}{2}\rfloor +ϵ)d^{n-(\lfloor \frac{n}{2}\rfloor +ϵ)}$ for each $ϵ=1,2,\dots ,n-\lfloor \frac{n}{2}\rfloor -1$. Clearly, the sequence $x_1,x_2,\dots ,x_{n-\lfloor \frac{n}{2}\rfloor -1}$ increases proportionally with the common ratio d, while $y_1,y_2,\dots ,y_{n-\lfloor \frac{n}{2}\rfloor -1}$ decreases with a ratio near $\frac{1}{d}$. Actually, $\frac{y_i}{y_{i-1}}=\frac{\lfloor \frac{n}{2}\rfloor +i}{\lfloor \frac{n}{2}\rfloor +(i-1)}\frac{1}{d}$. Three cases can be considered depending on the values of $x_{ϵ}$ and $y_{ϵ}$:

• $Case$ 1: $x_1\ge y_1$. The difference of $x_1$ and $y_1$ is the minimum among all differences of $x_i$ and $y_i$. Then, $\mathrm{rcn}=x_1$; thus, $\mathrm{rc}\left({\mathrm{WKP}}_{d,n}\right)\le d^{\lfloor \frac{n}{2}\rfloor }$ for the case $d^{\lfloor \frac{n}{2}\rfloor }\ge (\lfloor \frac{n}{2}\rfloor +1)d^{n-(\lfloor \frac{n}{2}\rfloor +1)}$ (see Table 3).

• $Case$ 2: $y_{n-\lfloor \frac{n}{2}\rfloor }\ge x_{n-\lfloor \frac{n}{2}\rfloor }$. The difference of $x_{n-\lfloor \frac{n}{2}\rfloor -1}$ and $y_{n-\lfloor \frac{n}{2}\rfloor -1}$ is the minimum among all differences of $x_i$ and $y_i$. Then, $\mathrm{rcn}=(n-1)d$; thus, $\mathrm{rc}\left({\mathrm{WKP}}_{d,n}\right)\le (n-1)d$ for the case $y_{n-\lfloor \frac{n}{2}\rfloor }\ge x_{n-\lfloor \frac{n}{2}\rfloor }$ (see Table 2).

• $Case$ 3: There exists an $ϵ$ such that $x_{ϵ}\ge y_{ϵ}$ and $x_{ϵ-1}\le y_{ϵ-1}$; then, $\mathrm{rcn}=\min (x_{ϵ},y_{ϵ-1})$, and an upper bound of $\mathrm{rc}\left({\mathrm{WKP}}_{d,n}\right)$ is established (see Table 3). □

In Table 1, the grey cell $x_1$ is the minimum of $x_1,x_2,x_3$, and $x_4$, while the grey cell $y_1$ is the maximum of $y_1,y_2,y_3$, and $y_4$. Obviously, the difference of $x_1=16,807$ and $y_1=14,406$ is the minimum among all differences of $x_i$ and $y_i$. Therefore, we obtain a rainbow $16,807$-coloring $c_p$ on ${\mathrm{WKP}}_{7,10}$ when $\lambda =\lfloor \frac{n}{2}\rfloor +1$. Table 2 and Table 3 provide the upper bounds of $\mathrm{rc}\left({\mathrm{WKP}}_{7,3}\right)$ and $\mathrm{rc}\left({\mathrm{WKP}}_{5,10}\right)$, respectively.

Table 3. The rainbow connection number (rcn) of ${\mathrm{WKP}}_{5,10}$ under $c_p$.

d = 5, n = 10
$ϵ$	1	2	3	4
$x_{ϵ}$	3125	15,625	78,125	390,625
$y_{ϵ}$	3750	875	200	45
rcn	3750

Table 2. The rainbow connection number (rcn) of ${\mathrm{WKP}}_{7,3}$ under $c_p$.

d = 7, n = 3
$ϵ$	1
$x_{ϵ}$	7
$y_{ϵ}$	14
rcn	14

Table 1. The rainbow connection number (rcn) of ${\mathrm{WKP}}_{7,10}$ under $c_p$.

d = 7, n = 10
$ϵ$	1	2	3	4
$x_{ϵ}$	16,807	117,649	823,543	5,764,801
$y_{ϵ}$	14,406	2401	392	63
rcn	16,807

4. Concluding Remarks

The WK-recursive pyramid ${\mathrm{WKP}}_{d,n}$ is structured upon the pyramid topology, comprising n layers of processing nodes. Each layer l of ${\mathrm{WKP}}_{d,n}$ is organized as a WK-recursive network ${\mathrm{WK}}_{d,l}$ with an expansion level of d times that of the layer directly above it. However, as the diameter of layer $L_n$ grows substantially for large values of d and n, it becomes imperative to partition $L_n$ into smaller WK-recursive subnetworks to achieve a reduced rainbow connection number. The expansion levels of these subnetworks serve as crucial parameters in algorithms aimed at determining rainbow connection numbers. In this paper, we undertake the task of determining the optimal expansion levels for these partitioned WK-recursive subnetworks, while additionally assessing whether a layer of ${\mathrm{WKP}}_{d,n}$ should undergo partitioning based on its specific characteristics and requirements.