The Equitable Coloring of Circulant Graphs

Abstract

A proper vertex coloring is equitable if the sizes of any two color classes differ by at most one. Any graph G with maximum degree $\Delta \left(G\right)$ admits an equitable $\left(\Delta \left(G\right)+1\right)$-coloring, computable in $O(\Delta \left(G\right)n^2)$ time for n vertices. A circulant graph $G(n;D)$ is the graph with vertex set ${\mathbb{Z}}_n$ and two vertices $x,y$ are adjacent if $|x-y|\in \pm Dmodn$. The partitioning problem in parallel decoding of multi-edge QC-LDPC codes can be interpreted as an equitable coloring problem. We prove some upper bounds for ${\chi }_{=}\left(G(n;D)\right)$ and develop equitable coloring algorithms, including pattern-based periodic coloring and step-based coloring. The proposed methods typically use fewer than $\Delta \left(G\right)+1$ colors and have computational complexity lower than $O(\Delta \left(G\right)n^2)$ for circulant graphs $G(n;D)$ with small $\left|D\right|$.

1. Introduction

As a variant of proper coloring, equitable coloring has attracted sustained attention due to both its theoretical significance and practical relevance. This concept was first introduced by Meyer [1]. A graph $G=(V,E)$ is said to be equitably m-colorable if its vertex set can be partitioned into m independent sets $V_1,\dots ,V_m$ such that $|\left|V_i\right|-|V_j||\le 1$ for all $i,j$. The minimum such m is called the equitable chromatic number and is denoted by ${\chi }_{=}\left(G\right)$. When $|V_i\left|=\right|V_j|$ for all $i,j$, the coloring is referred to as a strong equitable coloring. Determining ${\chi }_{=}\left(G\right)$ is generally NP-hard, which motivates research into special graph classes that admit structural characterizations or polynomial-time algorithms.

Equitable coloring naturally models scheduling and resource-allocation problems where tasks must be assigned to nearly balanced conflict-free groups. Applications arise in examination scheduling, register allocation, and related combinatorial optimization problems [2,3,4,5,6]. Circulant graphs constitute an important and highly symmetric class of graphs. They appear in coding theory and communication systems, particularly in the structural analysis of quasi-cyclic LDPC codes. In 5th Generation (5G) Mobile Network systems, parallel decoding is essential for improving throughput and reducing latency. When applied to parallel decoding, the division of a row in the base graph for multi-edge QC-LDPC codes can be modeled as an equitable coloring problem on $G(n;D)$. While the chromatic number of circulant graphs has been extensively investigated [7], their equitable chromatic number ${\chi }_{=}\left(G\right)$ remains less understood. In Reference [8], ${\chi }_{=}\left(G\right)$ was analyzed for circulant graphs with $\Delta \left(G\right)\le 4$. However, this work did not investigate ${\chi }_{=}\left(G\right)$ or equitable coloring schemes for circulant graphs with more general degree distributions. Building on this line of research, the present study continues to exploit the structural properties of circulant graphs and further investigates upper bounds for ${\chi }_{=}\left(G\right)$, together with corresponding equitable coloring schemes, for a broader class of circulant graphs.

The graph G supports an equitable m-coloring if $m>\Delta \left(G\right)$ [9]. Meyer [1] further conjectured that, with the exceptions of $K_n$ and $C_{2k+1}$, any graph G satisfies the inequality ${\chi }_{=}\left(G\right)\le \Delta \left(G\right)$. Substantial progress has been made for specific graph classes [10,11,12,13]. For instance, Chen and Lih  computed ${\chi }_{=}\left(G\right)$ for trees G, while Chen [14] characterized ${\chi }_{=}\left(G\right)$ for graphs with $\Delta \left(G\right)\le 3$ and chromatic number 3. Meyer’s conjecture was confirmed for bipartite graphs by Lih and Wu [15], and additional results concerning equitable coloring in $\Delta \left(G\right)$-colorable graphs were achieved by Kierstead and Kostochka [16].

From an algorithmic perspective, Kierstead [17] showed that for any $k\ge \Delta \left(G\right)+1$, an equitable k-coloring of an n-vertex graph can be computed in $O(\Delta \left(G\right)n^2)$ time. However, for structured graph classes such as circulant graphs, sharper bounds on ${\chi }_{=}\left(G\right)$ and more efficient coloring algorithms are still desirable. Below, we give the definition of circulant graphs and the properties of their adjacency matrices.

Definition 1.

Let $G(n;D)$ denote a circulant graph, where the vertex set is ${\mathbb{Z}}_n$ and the set of edges is defined as $E_n=\{(s,s+t)modn\mid t\in D,s\in {\mathbb{Z}}_n\},$ with $D\subset \{1,2,\dots ,\lfloor n/2\rfloor \}$ referred to as the connection set. Let $D=\{d_1,\dots ,d_{\mathit{max}}\}$, where $d_{\mathit{max}}$ is the largest element in D. Its adjacency matrix is a circulant matrix.

Let A be an $n\times n$ circulant matrix whose first row is $(a_0,a_1,\dots ,a_{n-1}),a_0=0,$ and

$$A[i,j]=a_{(j-i)modn}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\pm (j-i)modn\in D,\hfill \\ 0,\hfill & \mathit{otherwise},\hfill \end{array}\right.\forall i,j\in \{0,1,\dots ,n-1\}.$$

The circulant matrix A satisfies $A[i,j]=A[j,i]$. Except for the first row, each row is derived by cyclically shifting the preceding row to the right. Specifically, we have $A[i,j]=A[0,(j-i)modn].$ Therefore, for the circulant matrix A, its first row satisfies $A[0,d]=A[0,n-d]$.

$$A=\left(\begin{array}{ccccc}0& a_1& a_2& \cdots & a_{n-1}\\ a_{n-1}& 0& a_1& \cdots & a_{n-2}\\ a_{n-2}& a_{n-1}& 0& \cdots & a_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_1& a_2& a_3& \cdots & 0\end{array}\right)={\displaystyle \sum _{t=1}^{n-1}}{a}_t{L}^t.$$

The matrix A can be expressed as a polynomial of the basic cyclic shift matrix L, where $L^0=I, L \mathrm{is} \mathrm{defined} \mathrm{as} L=\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ 1& 0& 0& \cdots & 0\end{array}\right).$

In this paper, we investigate the equitable coloring of circulant graphs, focusing on establishing upper bounds for the equitable chromatic number and developing efficient algorithms. The paper is organized into three parts. In the first part, we review the research background and summarize relevant results on equitable coloring, including the definition of circulant graphs. The second part describes structural properties and periodic equitable coloring patterns, and illustrates several equitable coloring schemes. Finally, the third part presents the equitable coloring algorithm based on steps.

2. Equitable Pattern Periodic Coloring for Circulant Graphs

For circulant graphs, we aim to leverage their intrinsic structural properties to achieve an equitable coloring. Specifically, we select a representative subgraph of the graph $G(n;D)$ and determine the coloring based on the properties of this subgraph, rather than processing the entire graph.

Theorem 1.

For a graph $G(n;D)$, if $(d_{\mathit{max}}+1)\mid n$, then ${\chi }_{=}\left(G\right)$ is bounded above by $d_{\mathit{max}}+1$.

Proof. 

The matrix A can be expressed as a polynomial of L, $A={\sum }_{t=1}^{n-1}{a}_t{L}^t,$ and $a_t=a_{n-t}$.

$$\begin{array}{cc}\hfill A& ={\displaystyle \sum _{t=1}^{\lfloor n/2\rfloor }}{a}_t{L}^t+{\displaystyle \sum _{t=\lfloor n/2\rfloor +1}^{n-1}}{a}_t{L}^t\hfill \\ \hfill & ={\displaystyle \sum _{t=1}^{\lfloor n/2\rfloor }}{a}_t{L}^t+{\displaystyle \sum _{t=\lfloor n/2\rfloor +1}^{n-1}}{a}_{n-t}{\left(L^{n-t}\right)}^{\mathsf{T}}\hfill \\ \hfill & ={\displaystyle \sum _{t=1}^{\lfloor n/2\rfloor }}{a}_t{L}^t+{\displaystyle \sum _{t=1}^{n-\lfloor n/2\rfloor -1}}{a}_t{\left(L^t\right)}^{\mathsf{T}}\hfill \\ \hfill & ={\displaystyle \sum _{t=1}^{\lfloor n/2\rfloor }}{a}_t{L}^t+{\displaystyle \sum _{t=1}^{\lfloor (n-1)/2\rfloor }}{a}_t{\left(L^t\right)}^{\mathsf{T}}\hfill \\ \hfill & =A_{\mathrm{local}}+A_{\mathrm{local}}^{\mathsf{T}}-\left((n+1)mod2\right)a_{\lfloor n/2\rfloor }{L}^{\lfloor n/2\rfloor },\hfill \end{array}$$

where $A_{\mathrm{local}}={\sum }_{t=1}^{\lfloor n/2\rfloor }{a}_t{L}^t$, $A_{\mathrm{local}}^T$ denotes the transpose of $A_{\mathrm{local}}$.

Therefore, the graph G can be determined solely by the nonzero elements of $A_{\mathrm{local}}$. Assuming that $d_{\max }<\lfloor {\displaystyle \frac{n}{2}}\rfloor$, the following example illustrates the matrices A, $A_{\mathrm{local}}$, and $A_{\mathrm{local}}^{\mathsf{T}}$.

Under the conditions $d_{max}<⌊{\displaystyle \frac{n}{2}}⌋$ and $a_t=a_{n-t}$, the matrix A admits the following representation.

$$\begin{array}{c}A=\left(\begin{array}{ccccccccc}0& a_1& a_2& \cdots & a_{d_{\max }}& \cdots & a_{\lfloor n/2\rfloor }& \cdots & a_{n-1}\\ a_{n-1}& 0& a_1& \cdots & a_{d_{\max }-1}& \cdots & a_{\lfloor n/2\rfloor -1}& \cdots & a_{n-2}\\ a_{n-2}& a_{n-1}& 0& \cdots & a_{d_{\max }-2}& \cdots & a_{\lfloor n/2\rfloor -2}& \cdots & a_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮& \ddots & ⋮& & ⋮\\ a_1& a_2& a_3& \cdots & a_{d_{\max }+1}& \cdots & a_{\lfloor n/2\rfloor +1}& \cdots & 0\end{array}\right)\\ =circ(0,a_1,a_2,\cdots ,a_{d_{\max }},\underset{n-2d_{max}-1}{\underbrace{0,\cdots ,0,}}{a}_{d_{\max }},\cdots ,a_2,a_1)=A_{\mathrm{local}}+A_{\mathrm{local}}^{\mathsf{T}}.\end{array}$$

$$A_{\mathrm{local}}=\left(\begin{array}{cccccccc}0& a_1& a_2& \cdots & a_{d_{max}}& 0& \cdots & 0\\ 0& 0& a_1& \cdots & a_{d_{max}-1}& a_{d_{max}}& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 0& 0& a_1& \cdots & a_{d_{max}}& 0\\ 0& \cdots & \cdots & 0& 0& a_1& \cdots & a_{d_{max}}\\ a_{d_{max}}& 0& \cdots & \cdots & 0& 0& \cdots & a_{d_{max}-1}\\ ⋮& \ddots & & & & & \ddots & ⋮\\ a_1& a_2& \cdots & a_{d_{max}}& 0& 0& \cdots & 0\end{array}\right).$$

$$A_{\mathrm{local}}^{\mathsf{T}}=\left(\begin{array}{cccccccc}0& 0& \cdots & 0& 0& a_{d_{max}}& \cdots & a_1\\ a_1& 0& \ddots & ⋮& ⋮& 0& \ddots & a_2\\ a_2& a_1& \ddots & 0& ⋮& ⋮& \ddots & ⋮\\ ⋮& ⋮& \ddots & 0& 0& ⋮& & a_{d_{max}}\\ a_{d_{max}}& a_{d_{max}-1}& \cdots & a_1& 0& 0& & 0\\ 0& a_{d_{max}}& \cdots & ⋮& a_1& 0& \ddots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & \ddots & 0\\ 0& 0& \cdots & a_{d_{max}}& a_{d_{max}-1}& \cdots & a_1& 0\end{array}\right).$$

If $(d_{max}+1)\mid n$, then there exists an equitable $(d_{max}+1)$-coloring. One explicit coloring scheme is defined as follows. For each vertex labeled i, assign $c\left(i\right)=imod(d_{max}+1),$ where the set of colors is indexed by $\{0,1,\dots ,d_{max}\}$.

For any nonzero position $(i,j)$ in $A_{\mathrm{local}}$, we have $1\le (j-i)modn\le d_{max}$ and $c\left(j\right)-c\left(i\right)\equiv j-i(mod(d_{max}+1)).$ Because $1\le (j-i)modn\le d_{max}$ and $(d_{max}+1)\mid n$, we have $(j-i)\not\equiv 0(mod(d_{max}+1))$. Therefore, $c\left(j\right)\ne c\left(i\right)$, and this scheme is a proper coloring. Moreover, the condition $(d_{max}+1)\mid n$ guarantees that each color class has size exactly ${\displaystyle \frac{n}{d_{max}+1}}$, so the coloring is equitable. The idea above is to treat the subgraph corresponding to the orange region of Figure 1 as the coloring unit. For example, $G(9;\{1,2\left\}\right)$ and $G(28;\{2,3\left\}\right)$ are shown in Figure 2. □

Figure 1. An illustration of a coloring unit based on $A_{\mathrm{local}}$.

Figure 2. Two circulant graphs. (a) $G(9;\{1,2\left\}\right)$, (b) $G(28;\{2,3\left\}\right)$. Colors and letters indicate different vertex types: blue (A), orange (B), yellow (C), purple (D).

Corollary 1.

Consider a graph $G(n;D)$, and let $T=min\left\{k\in \mathbb{N}\mid k\mid n,k\ge d_{\max }+1\right\}$. Then, ${\chi }_{=}\left(G\right)\le T.$ The equitable coloring scheme is given by $c\left(i\right)=imodT$ for the vertex with label i, as shown in Figure 3.

Figure 3. $G(8;\{1,2\left\}\right)$. Colors and letters indicate different vertex types: blue (A), orange (B), yellow (C).

For the case where $(d_{\max }+1)\nmid n$, we follow a similar approach to find an equitable k-coloring with $d_{\max }+1\le k\le T$, as shown in Algorithm 1. We can enumerate all feasible assignments of the k colors to the vertices $i=kq,\dots ,n-1$, under the constraint that each of the k colors is used at most once in the current assignment. To carry out this enumeration, we employ a DSATUR-based algorithm. At each step, the algorithm gives priority to the uncolored vertex with the highest saturation degree. Specifically, for each uncolored vertex v, we first compute its saturation degree $sat\left(v\right)$, namely the number of distinct colors appearing in its already colored neighbors. An uncolored vertex with the maximum saturation degree is then selected. If several vertices attain the same saturation degree, the vertex with the maximum degree in the subgraph induced by the remaining uncolored vertices is chosen. If multiple vertices attain the maximum degree in the subgraph induced by the remaining uncolored vertices, the vertex with the smallest index is selected. After that, an admissible color is assigned to the selected vertex, and all possible choices are recursively enumerated. Once the vertex is colored, the saturation degrees of its uncolored neighbors are updated accordingly.

For a fixed k, let $r=nmodk$. The initial periodic coloring of the first $k\lfloor n/k\rfloor$ vertices takes $O\left(n\right)$ time. The subsequent DSATUR-based backtracking only processes the remaining r vertices. Since each of the k colors can be used at most once on these r vertices, the search tree has depth r and at most ${\displaystyle \frac{k!}{(k-r)!}}$ leaves. With incremental maintenance of saturation and forbidden color sets, each search node can be processed in $O(r+\Delta )$ time, where $\Delta$ is the maximum degree of $G(n;D)$. Hence, for a fixed k, the time complexity is $O\left(n+r\Delta +{\displaystyle \frac{k!}{(k-r)!}}(r+\Delta )\right).$ Therefore, the overall worst case complexity of Algorithm 1 is

$$O\left({\displaystyle \sum _{k=d_{max}+1}^T}\left[n+r\Delta +{\displaystyle \frac{k!}{(k-r)!}}(r+\Delta )\right]\right),r=nmodk.$$

In particular, the algorithm is exponential in the worst case. Since the equitable coloring problem is NP-hard, it is acceptable for the algorithm to have an exponential worst case running time.

After computing a coloring scheme for a symmetric unit, this scheme can be periodically replicated across other symmetric units, thereby achieving equitable coloring over the entire graph. This method is particularly suitable for graphs with complete symmetry or cyclic symmetry.

Algorithm 1: Equitable Coloring by Enumeration

Theorem 2.

Let $G=G(n;D)$, where $D\subseteq \{1,2,\dots ,d_{max}\}$, and assume that $2d_{max}\mid n.$ Let $G\left[\{0,1,\dots ,2d_{max}-1\}\right]$ be the induced subgraph on the first $2d_{max}$ vertices. Suppose that $G\left[\{0,1,\dots ,2d_{max}-1\}\right]$ admits a strong equitable coloring $c^{\prime }$ such that $\{c^{\prime }\left(0\right),c^{\prime }\left(1\right),\dots ,c^{\prime }(d_{max}-1)\}\cap \{c^{\prime }\left(d_{max}\right),c^{\prime }(d_{max}+1),\dots ,c^{\prime }(2d_{max}-1)\}=⌀.$ Define $c\left(v\right)=c^{\prime }(vmod2d_{max}),$ for all $v\in V\left(G\right).$ Then c is a strong equitable coloring of G.

Proof. 

Set $P:=2d_{max},Q:={\displaystyle \frac{n}{P}}.$ For each $t\in \{0,1,\dots ,Q-1\}$, define $B_t:=\{tP,tP+1,\dots ,tP+P-1\}(modn),$ and $L_t:=\{tP,tP+1,\dots ,tP+d_{max}-1\}(modn),$ $R_t:=\{tP+d_{max},tP+d_{max}+1,\dots ,tP+P-1\}(modn).$ Then $V\left(G\right)={⨆}_{t=0}^{Q-1}{B}_t,B_t=L_t\bigsqcup R_t.$

We first prove that c is a proper coloring.

For each t, consider the translation ${\tau }_t:\{0,1,\dots ,P-1\}\to B_t,{\tau }_t\left(i\right)=i+tP(modn).$ Since $G(n;D)$ is circulant, ${\tau }_t$ is an isomorphism from $G\left[B_0\right]=G\left[\{0,1,\dots ,P-1\}\right]$ to $G\left[B_t\right]$, as adjacency depends only on differences modulo n, which are preserved under translation. Moreover, by the definition of c, $c\left({\tau }_t\left(i\right)\right)=c^{\prime }\left(i\right),i=0,1,\dots ,P-1.$ Hence, the coloring of $G\left[B_t\right]$ is exactly a copy of the coloring $c^{\prime }$ of $G\left[B_0\right]$.

Let $xy\in E\left(G\right)$ with $x\in B_t$ and $y\in B_s$, where $t\ne s$, $t,s\in \{0,1,\dots ,Q-1\}$. Since $G(n;D)$ with $D\subseteq \{1,\dots ,d_{max}\}$, the distance between any two adjacent vertices is at most $d_{max}$. Therefore $s\equiv t\pm 1(modQ)$, because any two vertices in $B_t$ and $B_s$ with $s\not\equiv t\pm 1(modQ)$ are separated by distance greater than $d_{max}$. By symmetry, it suffices to consider an edge with $x\in B_t,y\in B_{(t+1)modQ}.$ Since $(y-x)modn\le d_{max}$, we must have $x\in R_t$ and $y\in L_{(t+1)modQ}$. By the periodic definition of c, all vertices in every $L_t$ receive colors from $\mathrm{Im}\left(c^{\prime }{|}_{\{0,1,\dots ,d_{max}-1\}}\right),$ while all vertices in every $R_t$ receive colors from $\mathrm{Im}\left(c^{\prime }{|}_{\{d_{max},d_{max}+1,\dots ,2d_{max}-1\}}\right).$ By definition, these two color sets are disjoint. Hence, for every edge $xy\in E\left(G\right)$ with $x\in R_t$ and $y\in L_{t+1}$, we have $c\left(x\right)\ne c\left(y\right).$

We conclude that c is a proper coloring of G.

Next, we prove that c is a strong equitable coloring. Since $c^{\prime }$ is a strong equitable coloring of $G\left[B_0\right]$, each color used in $G\left[B_0\right]$ appears the same number of times in the set $B_0=\{0,1,\dots ,P-1\}$. Because c is obtained by repeating this coloring on each of the Q sets $B_0,B_1,\dots ,B_{Q-1}$, every color class in G has size exactly Q times its size in $G\left[B_0\right]$. Hence, all color classes in G have equal cardinality.

Therefore, c is a strong equitable coloring of G. □

An illustration of a coloring unit based on $A_{\mathrm{local}}$ is shown in Figure 4. Consider the subgraph $G\left[B_0\right]$ and the coloring $c^{\prime }$. Since $\{c^{\prime }\left(0\right),c^{\prime }\left(1\right),\dots ,c^{\prime }(d_{max}-1)\}\cap \{c^{\prime }\left(d_{max}\right),c^{\prime }(d_{max}+1),\dots ,c^{\prime }(2d_{max}-1)\}=⌀$, the coloring unit can essentially be defined as the set of all edges induced by the nonzero elements in the orange triangular region, together with their associated vertices.

Figure 4. An illustration of a coloring unit triangular based on $A_{\mathrm{local}}$.

Definition 2.

Let $X=(x_1,x_2,\dots ,x_k)\in {\mathcal{C}}_1^k\mathit{and}Y=(y_1,y_2,\dots ,y_k)\in {\mathcal{C}}_2^k$ be two color sequences of the same length k, where ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are two sets of colors. We say that X and Y are coloring-pattern isomorphic if there exists a bijection $\phi :\mathrm{Im}\left(X\right)\to \mathrm{Im}\left(Y\right)$ such that $y_i=\phi \left(x_i\right),i=1,2,\dots ,k.$

Equivalently, $x_i=x_j⟺y_i=y_j,\forall i,j\in \{1,2,\dots ,k\}.$ In this case, we also say that X and Y have the same coloring pattern.

Moreover, if ${\mathcal{C}}_1\cap {\mathcal{C}}_2=⌀$, then X and Y are said to be pattern-isomorphic over disjoint color sets, also called disjoint-pattern isomorphic.

Remark 1.

For a graph H, a strong equitable k-coloring coincides with an equitable k-coloring under the divisibility condition $k\mid \left|V\right(H\left)\right|$. Therefore, it suffices to test only those values of k satisfying $k\mid \left|V\right(H\left)\right|$, and then decide whether H admits an equitable k-coloring. There exist many algorithms for equitable coloring, including the exact DSATUR-based algorithm of [18], the ILP-based algorithm of [19], and the polynomial-time algorithm of [17] for equitable k-coloring of graphs with maximum degree Δ when $k\ge \Delta +1$.

Let $L_0=\{0,1,\dots ,d_{max}-1\}$ and $R_0=\{d_{max},d_{max}+1,\dots ,2d_{max}-1\}$. To construct a strong equitable coloring $c^{\prime }$ of $G\left[B_0\right]$, one may directly color $G\left[L_0\right]$ and $G\left[R_0\right]$ with disjoint color sets. The most intuitive approach is to require that the two color sequences $\left(c^{\prime }\left(0\right),c^{\prime }\left(1\right),\dots ,c^{\prime }(d_{max}-1)\right)$ and $\left(c^{\prime }\left(d_{max}\right),c^{\prime }(d_{max}+1),\dots ,c^{\prime }(2d_{max}-1)\right)$ are pattern-isomorphic over disjoint color sets. In this case, it suffices to consider only divisors $k\mid d_{max}$, find a strong equitable k-coloring of $G\left[L_0\right]$, and then assign an isomorphic coloring pattern to $G\left[R_0\right]$ using a disjoint set of k colors. In the following corollary, we use the algorithm of [17] as a concrete method for finding a strong equitable k-coloring of H.

As an example, different isomorphic coloring patterns of strong equitable colorings are illustrated for the induced subgraph $G\left[B_0\right]=G\left[\{0,1,\dots ,2d_{max}-1\}\right]$ of the circulant graph $G(16;\{3,4\left\}\right)$—see Figure 5.

Figure 5. The example of $G(16;\{3,4\left\}\right)$ with different isomorphic coloring patterns. (a) $c^{\prime }\left(G\left[B_0\right]\right)=\{A,B,A,B;C,D,C,D\}$ of $G(16;\{3,4\left\}\right)$, (b) $c^{\prime }\left(G\left[B_0\right]\right)=\{A,A,B,B;C,C,D,D\}$ of $G(16;\{3,4\left\}\right)$. Colors and letters indicate different vertex types: blue (A), orange (B), yellow (C), purple (D).

Corollary 2.

Given the circulant graph $G(n;D)$, we can find the smallest integer $d\ge d_{max}$ such that $2d\mid n$. The equitable coloring algorithm is given in Algorithm 2. Let $F:=min\{2d\mid d\in \mathbb{Z},d\ge d_{max},2d\mid n\}$; then, ${\chi }_{=}\left(G\right)\le F.$

Algorithm 2: Strong Equitable Coloring by Symmetric Subgraph

In Algorithm 2, the graph $H=G\left[\right\{0,1,\dots ,d-1\left\}\right]$ is first colored by an equitable k-coloring algorithm, where k is chosen such that $k\ge \Delta \left(H\right)+1\mathrm{and}k\mid d.$ Since $k\mid d=\left|V\right(H\left)\right|$, the resulting coloring is actually a strong equitable coloring. The coloring on $G\left[\right\{d,d+1,\dots ,2d-1\left\}\right]$ is then obtained by a bijective relabeling of the colors, namely $c^{\prime }(d+i)=f\left(i\right)+k$ for $i=0,1,\dots ,d-1$. Hence, the two color sequences are pattern-isomorphic, and their color sets are disjoint. Finally, extend $c^{\prime }$ periodically to all vertices of G with period $2d$. By Theorem 2, the same argument shows that $G(n;D)$ admits a strong equitable coloring.

Assume that arithmetic operations on indices are performed in constant time. The time complexity of Algorithm 2 is dominated by the equitable coloring step on the graph $H=G\left[\right\{0,1,\dots ,d-1\left\}\right]$. Recovering D, searching for the smallest integer $d\ge d_{max}$ such that $2d\mid n$, computing $\Delta \left(H\right)$, choosing k, and performing the periodic extension take $O\left(n\right)$ time. Constructing the adjacency matrix of H and computing all degrees in H take $O\left(d^2\right)$ time. Therefore, these preprocessing and extension steps together require $O(n+d^2)$ time. If the equitable k-coloring of H is computed by the algorithm from [17], its running time is $O(\Delta \left(H\right)d^2)$. Therefore, the overall complexity is $O(\Delta \left(H\right)d^2+n),$ which is $O\left(\right|D|d^2+n)$ in the worst case.

When $2d_{max}\nmid n$, let $F:=min\{2d\mid d\in \mathbb{Z},d\ge d_{max},2d\mid n\}$. For any even number $l_B$ with $2d_{max}\le l_B\le F$, the same construction can be applied to the induced subgraph $G\left[\{0,1,\dots ,l_B-1\}\right]$. More precisely, the vertex set is divided into two equal halves, and a strong equitable coloring is constructed on these two halves using disjoint color sets. The resulting coloring pattern is then extended to the first $l_B\lfloor n/l_B\rfloor$ vertices of $G(n;D)$. For the remaining $nmod{l}_B$ vertices, the colors are assigned by enumeration, where each color is used at most once among the residual vertices. This enumeration can be implemented in the same way as Algorithm 1.

Theorem 3.

Let $G=G(n;D)$ be a circulant graph with $D\subseteq \{1,2,\dots ,\lfloor n/2\rfloor \}$, and let $d_{max}=maxD$. Define $T=min\{k\in \mathbb{N}\mid k\mid n,k\ge d_{max}+1\}$ and $F:=min\{2d\mid d\in \mathbb{Z},d\ge d_{max},2d\mid n\}$. Then the equitable chromatic number of G satisfies ${\chi }_{=}\left(G\right)\le min\{T,F\}$.

In particular, if $D=\{1,2,\dots ,m\}$, then $m+1\mid n$ implies ${\chi }_{=}\left(G\right)=m+1$, whereas $m+1\nmid n$ implies ${\chi }_{=}\left(G\right)\ge m+2$. Consequently, for $D=\{1,2\}$, if $3\mid n$, then ${\chi }_{=}\left(G\right)=3$; if $3\nmid n$, then ${\chi }_{=}\left(G\right)\ge 4$.

Proof. 

By Corollaries 1 and 2, the graph $G(n;D)$ admits an equitable coloring with T colors and also admits an equitable coloring with F colors. Therefore, ${\chi }_{=}\left(G\right)\le min\{T,F\}$.

Now consider the special case $D=\{1,2,\dots ,m\}$. For any $j\in {\mathbb{Z}}_n$, the set $\{j,j+1,\dots ,j+m\}(modn)$ contains $m+1$ vertices, and for any two distinct vertices $u,v$ in this set, $u-v\in \pm D(modn)$. Hence this set induces a complete graph $K_{m+1}$. Therefore, every proper coloring of G requires at least $m+1$ colors, that is, ${\chi }_{=}\left(G\right)\ge m+1$.

If $m+1\mid n$, then $m+1$ is an admissible divisor in the definition of T, since $d_{max}=m$. Thus $T=m+1$, and by the first part of the theorem, ${\chi }_{=}\left(G\right)\le m+1$. Combining this with ${\chi }_{=}\left(G\right)\ge m+1$, we obtain ${\chi }_{=}\left(G\right)=m+1$.

It remains to show that if $m+1\nmid n$, then no proper $(m+1)$-coloring exists. Suppose, to the contrary, that G admits a proper coloring with exactly $m+1$ colors. Since every set of $m+1$ consecutive vertices induces a clique $K_{m+1}$, the vertices $j,j+1,\dots ,j+m$ must receive all $m+1$ colors. The next set $\{j+1,j+2,\dots ,j+m+1\}$ also induces a clique $K_{m+1}$ and must again receive all $m+1$ colors. Since these two sets share the m vertices $j+1,\dots ,j+m$, the only possible color for $j+m+1$ is the color assigned to j. Hence the coloring must satisfy $c(j+m+1)=c\left(j\right)$ for all $j\in {\mathbb{Z}}_n$. Thus the coloring is $(m+1)$-periodic.

Such a periodic coloring is compatible with the cyclic vertex set ${\mathbb{Z}}_n$ only if $m+1\mid n$. If $m+1\nmid n$, the periodicity condition cannot be maintained consistently. Equivalently, it would force two vertices in some set of $m+1$ consecutive vertices to receive the same color, contradicting the fact that every such set induces a clique $K_{m+1}$. Therefore, when $m+1\nmid n$, no proper $(m+1)$-coloring exists, and hence ${\chi }_{=}\left(G\right)\ge m+2$.

For $D=\{1,2\}$, we have $m=2$. Therefore, if $3\mid n$, then ${\chi }_{=}\left(G\right)=3$. If $3\nmid n$, then ${\chi }_{=}\left(G\right)\ge 4$. This completes the proof. □

3. Equitable Coloring Based on Steps for Circulant Graphs

The following section outlines an algorithm based on steps for finding an equitable coloring of the circulant graph G and analyzes the effectiveness of the algorithm.

A 2-regular circulant graph $G(n;D)$ where $\left|D\right|=1$ can be represented as a collection of $gcd(d,n)$ cycles, each of which has length ${\displaystyle \frac{n}{gcd(d,n)}}$. Therefore, the equitable chromatic number of $G(n;D)$ is 2 or 3. Based on the above property, we propose the following coloring algorithm.

Lemma 1.

Let $D=\{d_1,d_2,\dots ,d_m\}$, and for each $i=1,2,\dots ,m$, let $g_i=gcd(n,d_i)$, $L_i=n/g_i$, and let $t_i$ be the smallest prime factor of $L_i$. For each vertex $x\in {\mathbb{Z}}_n$, uniquely write $x\equiv a_i+(s_i+t_i{k}_i)d_i(modn)$, where $0\le a_i\le g_i-1$, $0\le s_i\le t_i-1$, and $0\le k_i\le L_i/t_i-1$. Define the i-th component coloring by $c_i\left(x\right)=(a_i+s_i)mod{t}_i$, and define the integrated coloring by $\mathsf{\Phi }\left(x\right)=(c_1\left(x\right),c_2\left(x\right),\dots ,c_m\left(x\right))$. Then Φ is a proper coloring of $G(n;D)$.

Proof. 

For each $i\in \{1,2,\dots ,m\}$, put ${\overline{d}}_i=d_i/g_i$. Since $g_i=gcd(n,d_i)$ and $L_i=n/g_i$, we have $gcd({\overline{d}}_i,L_i)=1$. Hence ${\overline{d}}_i$ is invertible modulo $L_i$.

For any $x\in {\mathbb{Z}}_n$, the residue $a_i\equiv x(mod{g}_i)$ with $0\le a_i\le g_i-1$ is uniquely determined. Since $x-a_i$ is divisible by $g_i$, there exists an integer $q_i$ such that $x-a_i=g_i{q}_i$. The congruence $x\equiv a_i+r_i{d}_i(modn)$ is equivalent to $q_i\equiv r_i{\overline{d}}_i(mod{L}_i)$. Since ${\overline{d}}_i$ is invertible modulo $L_i$, the value of $r_i$ is uniquely determined modulo $L_i$. Taking the representative $0\le r_i\le L_i-1$, and using $t_i\mid L_i$, we can write $r_i=s_i+t_i{k}_i$ uniquely with $0\le s_i\le t_i-1$ and $0\le k_i\le L_i/t_i-1$. Therefore $s_i$ is uniquely determined by x, and the component coloring $c_i\left(x\right)=(a_i+s_i)mod{t}_i$ is well defined.

It remains to prove that $\mathsf{\Phi }$ is proper. Let $x,y\in {\mathbb{Z}}_n$ be adjacent in $G(n;D)$. Then there exist $j\in \{1,2,\dots ,m\}$ and $\epsilon \in \{1,-1\}$ such that $y\equiv x+\epsilon d_j(modn)$. Write $x\equiv a_j+(s_j+t_j{k}_j)d_j(modn)$. Since $g_j\mid d_j$, adding $\epsilon d_j$ does not change the residue modulo $g_j$. Hence the corresponding residue for y is still $a_j$. Moreover, $y\equiv a_j+(s_j+t_j{k}_j+\epsilon )d_j(modn)$. Therefore, in the j-th component, the parameter $s_j$ is replaced by $s_j+\epsilon$ modulo $t_j$. Hence $c_j\left(y\right)\equiv a_j+s_j+\epsilon (mod{t}_j)$, while $c_j\left(x\right)\equiv a_j+s_j(mod{t}_j)$. Consequently, $c_j\left(y\right)-c_j\left(x\right)\equiv \epsilon (mod{t}_j)$. Since $t_j$ is a prime factor of $L_j$, we have $t_j\ge 2$. Thus $\epsilon \not\equiv 0(mod{t}_j)$, and so $c_j\left(y\right)\ne c_j\left(x\right)$. Hence the j-th coordinates of $\mathsf{\Phi }\left(x\right)$ and $\mathsf{\Phi }\left(y\right)$ are different, which implies $\mathsf{\Phi }\left(x\right)\ne \mathsf{\Phi }\left(y\right)$. Therefore every pair of adjacent vertices receives distinct integrated colors, and $\mathsf{\Phi }$ is a proper coloring of $G(n;D)$.

This Lemma can be understood from the structural viewpoint of circulant graphs. For each fixed i, the coloring $c_i$ is a strong equitable coloring of the circulant graph $G(n;\left\{d_i\right\})$. Since $g_i=gcd(n,d_i)$, the graph $G(n;\left\{d_i\right\})$ consists of $g_i$ disjoint cycles, each of length $L_i=n/g_i$. On every such cycle, $a_i$ is fixed, while moving from a vertex x to an adjacent vertex y, then $y\equiv x\pm d_i(modn)$. Thus, the parameter $s_i$ changes to $s_i\pm 1$ modulo $t_i$, whereas $a_i$ is fixed. Consequently, $c_i\left(y\right)-c_i\left(x\right)\equiv \pm 1(mod{t}_i)$. Since $t_i\ge 2$, this implies $c_i\left(y\right)\ne c_i\left(x\right)$. Hence $c_i$ is a proper coloring. Since $t_i\mid L_i$, the coloring pattern on each cycle of length $L_i$ is $t_i$-periodic. More precisely, the vertices in each component of $G(n;\left\{d_i\right\})$ can be written as $a_i+r_i{d}_i$ with $0\le r_i\le L_i-1$, where $a_i\in \{0,1,\dots ,g_i-1\}$. Writing $r_i=s_i+t_i{k}_i$ with $0\le s_i\le t_i-1$, the coloring $c_i\left(x\right)=(a_i+s_i)mod{t}_i$ assigns colors periodically with period $t_i$ along the cycle. Moreover, since $L_i$ is divisible by $t_i$, each color appears exactly $L_i/t_i$ times on every cycle. Since $G(n;\left\{d_i\right\})$ consists of $g_i$ disjoint cycles of length $L_i$, each color appears exactly $g_i{L}_i/t_i=n/t_i$ times in the whole graph. Therefore $c_i$ is a strong equitable coloring of the circulant graph $G(n;\left\{d_i\right\})$ with exactly $t_i$ colors.

Now let $x,y\in {\mathbb{Z}}_n$ be adjacent in $G(n;D)$. By the definition of a circulant graph, there exists some $d_r\in D$ such that $y\equiv x\pm d_r(modn)$. Hence x and y are adjacent in the subgraph $G(n;\left\{d_r\right\})$, and it follows that $c_r\left(x\right)\ne c_r\left(y\right)$. Therefore the two integrated color vectors differ in their rth coordinate, that is, $\mathsf{\Phi }\left(x\right)\ne \mathsf{\Phi }\left(y\right)$. Thus $\mathsf{\Phi }$ is a proper coloring of $G(n;D)$. □

Lemma 2.

Set $L=lcm(g_1{t}_1,g_2{t}_2,\dots ,g_m{t}_m)$. Then L is a period of Φ, that is, $\mathsf{\Phi }(x+L)=\mathsf{\Phi }\left(x\right)$ for all $x\in {\mathbb{Z}}_n$. Consequently, if K denotes the number of colors used by Φ, then $K=\left|\left\{\mathsf{\Phi }\right(x):x=0,1,\dots ,L-1\}\right|$, and in particular $K\le min\left\{L,{\prod }_{i=1}^m{t}_i\right\}$.

Proof. 

For each $i\in \{1,2,\dots ,m\}$, write $d_i=g_i{\overline{d}}_i$ and $n=g_i{L}_i$. Then $gcd({\overline{d}}_i,L_i)=1$. We first prove that $g_i{t}_i$ is a period of $c_i$. Let $x^{\prime }=x+g_i{t}_i$; since $x^{\prime }\equiv x(mod{g}_i)$, we have $a_i\left(x^{\prime }\right)=a_i\left(x\right)$. Moreover, if $x\equiv a_i+r_i{d}_i(modn)$, then after dividing by $g_i$, we obtain $(x-a_i)/g_i\equiv r_i{\overline{d}}_i(mod{L}_i)$. Since ${\overline{d}}_i$ is invertible modulo $L_i$, it is also invertible modulo $t_i$, and hence $r_i\equiv {\overline{d}}_i^{-1}(x-a_i)/g_i(mod{t}_i)$. For $x^{\prime }=x+g_i{t}_i$, we have $(x^{\prime }-a_i)/g_i=(x-a_i)/g_i+t_i$. Therefore, $r_i\left(x^{\prime }\right)\equiv {\overline{d}}_i^{-1}((x-a_i)/g_i+t_i)\equiv {\overline{d}}_i^{-1}(x-a_i)/g_i\equiv r_i\left(x\right)(mod{t}_i)$. Thus $s_i\left(x^{\prime }\right)=s_i\left(x\right)$. Together with $a_i\left(x^{\prime }\right)=a_i\left(x\right)$, this gives $c_i(x+g_i{t}_i)=c_i\left(x\right)$. Hence $g_i{t}_i$ is a period of $c_i$.

Since $L=lcm(g_1{t}_1,g_2{t}_2,\dots ,g_m{t}_m)$, we have $g_i{t}_i\mid L$ for every i. Therefore $c_i(x+L)=c_i\left(x\right)$ for all $i=1,2,\dots ,m$. Consequently, $\mathsf{\Phi }(x+L)=(c_1(x+L),c_2(x+L),\dots ,c_m(x+L))=(c_1\left(x\right),c_2\left(x\right),\dots ,c_m\left(x\right))=\mathsf{\Phi }\left(x\right)$. Hence L is a period of the integrated coloring $\mathsf{\Phi }$.

Since $\mathsf{\Phi }$ is periodic with period L, all colors used by $\mathsf{\Phi }$ appear within one period. Therefore $K=\left|\left\{\mathsf{\Phi }\right(x):x=0,1,\dots ,L-1\}\right|$, and $K\le L$. On the other hand, each component $c_i$ takes values in ${\mathbb{Z}}_{t_i}$, so $\mathsf{\Phi }\left(x\right)\in {\mathbb{Z}}_{t_1}\times \cdots \times {\mathbb{Z}}_{t_m}$. Hence $K\le {\prod }_{i=1}^m{t}_i$. Therefore, the number of colors satisfies $K\le min\left\{L,{\prod }_{i=1}^m{t}_i\right\}$. □

Remark 2.

Two sufficient conditions under which Φ can yield an equitable coloring are given as follows.

1. 

If $d_i\equiv 1(mod{t}_i)$ for every $i\in \{1,2,\dots ,m\}$, then $\mathsf{\Phi }\left(x\right)=(xmod{t}_1,\dots ,xmod{t}_m)$. Writing $\lambda =lcm(t_1,\dots ,t_m)$, one has $K=|Im(\mathsf{\Phi }\left)\right|=\lambda$, and each color class has size $n/\lambda$.

2. 

If $N_i:=g_i{t}_i$ satisfies $gcd(N_i,N_j)=1$ for all $i\ne j$, then every vector color occurs, $K={\prod }_{i=1}^m{t}_i$, and each color class has size $n/{\prod }_{i=1}^m{t}_i$.

Proof. 

By the definition of the stepwise coloring, every $x\in {\mathbb{Z}}_n$ can be written as $x\equiv a_i+s_i{d}_i+k{t}_i{d}_i(modn)$, where $0\le a_i\le g_i-1$, $0\le s_i\le t_i-1$, and $c_i\left(x\right)\equiv a_i+s_i(mod{t}_i)$. Since $d_i\equiv 1(mod{t}_i)$, reducing modulo $t_i$ gives $x\equiv a_i+s_i(mod{t}_i)$, hence $c_i\left(x\right)\equiv x(mod{t}_i)$. Therefore $\mathsf{\Phi }\left(x\right)=(xmod{t}_1,\dots ,xmod{t}_m)$. Thus $\mathsf{\Phi }$ is a homomorphism, $ker\left(\mathsf{\Phi }\right)=\{x\in {\mathbb{Z}}_n:x\equiv 0(mod{t}_i)\mathrm{for}\mathrm{any}i\}$, and so $|ker(\mathsf{\Phi }\left)\right|=n/\lambda$, where $\lambda =lcm(t_1,\dots ,t_m)$. By the first isomorphism theorem for finite groups, $K=|Im\left(\mathsf{\Phi }\right)|=|{\mathbb{Z}}_n|/|ker\left(\mathsf{\Phi }\right)|=\lambda$. Since every fiber of a homomorphism is a coset of the kernel, every color class has size $n/\lambda$. Hence the integrated coloring is strictly equitable.

Since $t_i\mid L_i=n/g_i$, $N_i=g_i{t}_i\mid n$ for every i. Under $gcd(N_i,N_j)=1$ for $i\ne j$, it follows that $M:=lcm(N_1,\dots ,N_m)={\prod }_{i=1}^m{N}_i$, hence $M\mid n$. For fixed i and $e_i\in {\mathbb{Z}}_{t_i}$, the stepwise coloring modulo $N_i$ consists of $g_i$ cycles of length $t_i$, so the color $e_i$ occurs exactly $g_i$ times modulo $N_i$. Denote by $E_{i,e_i}\subseteq {\mathbb{Z}}_{N_i}$ the corresponding residue set, then $|E_{i,e_i}|=g_i$. Now let $\mathbf{r}=(r_1,\dots ,r_m)\in {\prod }_{i=1}^m{\mathbb{Z}}_{t_i}$. For any choice $u_i\in E_{i,e_i}$, the Chinese Remainder Theorem yields a unique $x(modM)$ satisfying $x\equiv u_i(mod{N}_i)$ for all i. Hence, in one period modulo M, the color $\mathbf{r}$ occurs exactly ${\prod }_{i=1}^m{g}_i$ times. Since $M\mid n$, each residue class modulo M is repeated $n/M$ times in ${\mathbb{Z}}_n$. Therefore each final color class has size $(n/M){\prod }_{i=1}^m{g}_i=n/{\prod }_{i=1}^m{t}_i$. Thus the integrated coloring is strictly equitable. Moreover, every vector color $\mathbf{r}\in {\prod }_{i=1}^m{\mathbb{Z}}_{t_i}$ occurs, so $K={\prod }_{i=1}^m{t}_i$. □

Theorem 4.

For a circulant graph $G(n;D)$, the equitable coloring algorithm-based steps are described as follows, as shown in Algorithm 3.

1. 

Input: D, $G(n;D)$.

2. 

Stepwise Coloring: For each $G(n;\left\{d_i\right\}),d_i\in D$, compute coloring independently.

For each $i=1,2,\dots ,m$, where $m=\left|D\right|$, compute $g_i=gcd(n,d_i)$ and $L_i={\displaystyle \frac{n}{g_i}}$, and let $t_i$ denote the smallest prime factor of $L_i$. Define a $t_i$-coloring $c_i$ as follows. For each vertex $x\in {\mathbb{Z}}_n$ satisfying

$$x\equiv a_i+(s_i+t_i{k}_i)d_i(modn),a_i=0,1,\dots ,g_i-1,s_i=0,1,\dots ,t_i-1,k_i\in \mathbb{Z},$$

assign $c_i\left(x\right)=(a_i+s_i)mod{t}_i.$

3. 

Integration: Save an m-dimensional color vector $\mathsf{\Phi }\left(x\right)$ for each vertex x, assign final colors by vector comparison.

$$\mathsf{\Phi }\left(x\right)=\left((a_1+s_1)mod{t}_1,\dots ,(a_i+s_i)mod{t}_i,\dots ,(a_m+s_m)mod{t}_m\right).$$

4. 

Verification: Check whether the coloring is equitable and whether the number of colors is less than or equal to $\Delta \left(G\right)+1$. If satisfied, output the coloring scheme, otherwise, apply the algorithm from [17] to obtain an equitable $(\Delta (G)+1)$-coloring.

5. 

Output: Equitable coloring scheme.

Proof. 

For each $G(n;\left\{d_i\right\}),d_i\in D$, the coloring is computed independently. An m-dimensional color vector $\mathsf{\Phi }\left(x\right)$ is stored for each vertex x, and the final colors are assigned through vector comparison. According to Lemma 1, this construction ensures a proper coloring of G. The verification guarantees the equitable property. □

Algorithm 3: Equitable Coloring Based on Steps for Circulant Graphs

Complexity analysis: Let $m=\left|D\right|$. We analyze the complexity of the algorithm in four parts.

First, the preprocessing stage computes $g_i=gcd(n,d_i)$, $L_i=n/g_i$, and the smallest prime factor $t_i$ of $L_i$ for each $d_i\in D$. The values $g_i=gcd(n,d_i)$ can be computed by the Euclidean algorithm, and each gcd computation takes $O(logn)$ time. Hence all gcd computations take $O(mlogn)$ time. For the computation of $t_i$, only the m integers $L_1,L_2,\dots ,L_m$ are used. If $t_i$ is computed directly by trial division for each $L_i$, then the worst case cost for one $L_i$ is $O\left(\sqrt{L_i}\right)\subseteq O\left(\sqrt{n}\right)$. Therefore, the total cost of computing all $t_i$’s is $O\left(m\sqrt{n}\right)$ in the worst case. Thus the preprocessing cost is $O(mlogn+m\sqrt{n})$.

Next, consider the pattern-coloring stage for a fixed $d_i$. The step $d_i$ decomposes the block into $g_i$ residue classes, each of length $L_i=\frac{n}{g_i}.$ For each $a_i\in \{0,\dots ,g_i-1\}$ and $s_i\in \{0,\dots ,t_i-1\}$, the vertices $(a_i+s_i{d}_i+k{t}_i{d}_i)modn$ are pairwise distinct for $k=0,1,\dots ,L_i/t_i-1.$ Therefore, the innermost loop executes exactly $L_i/t_i$ times for each pair $(a_i,s_i)$. Summing over all $a_i$ and $s_i$, the total number of assignments for this fixed $d_i$ is

$${\displaystyle \sum _{a_i=0}^{g_i-1}}{\displaystyle \sum _{s_i=0}^{t_i-1}}{\displaystyle \frac{L_i}{t_i}}=g_i{t}_i{\displaystyle \frac{L_i}{t_i}}=g_i{L}_i=n.$$

Thus the total cost of the pattern-coloring stage over all $d_i\in D$ is $\mathsf{\Theta }\left(mn\right).$

For the integration stage, each vertex x has an m dimensional color vector $\mathsf{\Phi }\left(x\right)=(c_1\left(x\right),\dots ,c_m\left(x\right)).$ Forming all coordinates over all vertices already accounts for $\mathsf{\Theta }\left(mn\right)$ work. To merge equal vectors deterministically, we lexicographically sort the n vectors by stable counting sort, processing the coordinates from right to left. Since the i-th coordinate takes values in $\{0,\dots ,t_i-1\}$, one pass costs $O(n+t_i)$. Therefore, the total sorting cost is $O\left({\sum }_{i=1}^m(n+t_i)\right)=O\left(mn\right),$ because $t_i\le L_i\le n$ for every i. A final linear scan of the sorted list assigns one merged color to each distinct vector, which costs $O\left(n\right)$. Hence the integration stage is $O\left(mn\right)$.

The verification stage consists of two checks. First, equitability is checked by counting the sizes of all color classes, which costs $O(n+K)$, where K is the number of colors produced. Second, verifying the bound $K\le \Delta \left(G\right)+1$ takes $O\left(1\right)$ time once K is known. Therefore, the overall verification cost is $O\left(n\right)$.

Combining the above bounds, the deterministic running time of the construction, integration and verification parts is $O(m\sqrt{n}+mlogn+mn)=O\left(mn\right).$

If the fallback algorithm is invoked, then the total running time becomes $O\left(mn+T_{\mathrm{K}}(n,\Delta \left(G\right),|E\left(G\right)\left|\right)\right),$ where $T_{\mathrm{K}}(n,\Delta \left(G\right),|E\left(G\right)\left|\right)$ denotes the running time of the general equitable coloring subroutine used in the fallback step. In particular, when the equitable coloring algorithm from [17] is used as the subroutine and implemented in $O(\Delta \left(G\right)n^2)$ time, the worst case running time is dominated by this subroutine. Therefore, the overall running time is $O(\Delta \left(G\right)n^2)$.

We first present two illustrative examples. The circulant graph $G(20;\{2,3,6\left\}\right)$ admits an equitable 4-coloring, and the circulant graph $G(22;\{1,3,5,7,9,11\left\}\right)$ admits an equitable 2—see Figure 6. These examples show that the step-based construction can produce equitable colorings with relatively few color classes for some instances.

Figure 6. Examples of circulant graphs with different parameters. (a) $G(20;\{2,3,6\left\}\right)$, (b) $G(22;\{1,3,5,7,9,11\left\}\right)$. Colors and letters indicate different vertex types: blue (A), orange (B), yellow (C), purple (D).

To evaluate the proposed method on circulant graphs, we conducted two groups of random experiments. In each parameter configuration, twenty independent circulant graphs were generated, where the step set D was sampled uniformly at random from $\{1,2,\dots ,\lfloor n/2\rfloor \}$ without repetition. For each instance, we recorded the number of colors produced by the step-based phase, whether this coloring was already equitable, the final number of colors returned by the complete step-based equitable coloring algorithm, and the MATLAB R2023a running time. In the first setting, either $n=128$ or $m=\left|D\right|=5$ was fixed. In the second setting, both n and $m=\left|D\right|$ were varied, and twenty random instances were generated for each $(n,m)$ pair.

The results are reported in Figure 7 and Figure 8. Figure 7 shows the average number of color classes under the fixed-parameter setting $n=128$ or $m=\left|D\right|=5$. The results indicate that the proposed method often returns colorings using fewer colors than the benchmark $\Delta \left(G\right)+1$. Moreover, in many tested cases, the step-based phase already produces an equitable coloring, so the fallback procedure is not required. Figure 8 presents the results for randomly selected $(n,m)$ pairs. The gray plane represents the benchmark value $\Delta \left(G\right)+1$. Scatter points below this plane indicate instances where the computed coloring uses fewer than $\Delta \left(G\right)+1$ colors.

Figure 7. Comparison of the average number of equitable colors under different parameters. (a) The average number of equitable colors, $n=128$, (b) The average number of equitable colors, $\left|D\right|=5$.

Figure 8. The average number of equitable colors.

We compare the obtained number of colors with $\Delta \left(G\right)+1$ because it is a classical universal upper bound benchmark for graph equitable coloring. Therefore, this comparison provides a simple reference for measuring how far the computed coloring lies below a general worst case guarantee.

However, $\Delta \left(G\right)+1$ is used only as a benchmark, not as the sole criterion for validating the proposed algorithm. For this reason, we also report the performance of the step-based phase as shown in Figure 9. The gap is defined as the difference between the number of colors obtained by the step-coloring stage alone and $\Delta \left(G\right)+1$. A positive frequency of $\mathrm{gap}<0$ indicates that the step-coloring component has an advantage, since it can produce a coloring using fewer than $\Delta \left(G\right)+1$ colors in some cases. In contrast, if the frequency of $\mathrm{gap}<0$ tends to zero, then the fallback algorithm is generally required to guarantee a smaller equitable coloring. As shown in Figure 9a, the proposed algorithm is more effective when m is relatively small. The standard deviation of the coloring numbers in the step-based equitable coloring is also evaluated. Figure 9b shows that the proposed method is relatively stable when m and n are small.

Figure 9. Performance of the step-based coloring phase under different parameters. (a) Frequency of $\mathrm{gap}<0$, where $\mathrm{gap}$ is defined as the difference between the number of colors obtained by the step-coloring stage alone and $\Delta \left(G\right)+1$, (b) standard deviation of coloring numbers in step-based equitable coloring.

To provide detailed computational evidence and strengthen the empirical evaluation, we compare three equitable coloring algorithms on the same set of test instances: the proposed step-based equitable coloring algorithm, the exact DSATUR-based algorithm of [18], the ILP-based algorithm of [19]. Specifically, we set $n\in \{10,15,20,25,30\}$. For each fixed n, the parameter m is selected as $m=2:2:\lfloor n/2\rfloor$. For each fixed pair $(n,m)$, 20 random subsets $D\subseteq \{1,\dots ,\lfloor n/2\rfloor \}$ with $\left|D\right|=m$ are independently generated. In each trial, the three algorithms are applied to exactly the same D, which ensures an equitable and reproducible comparison.

The DSATUR-based algorithm is implemented as an exact feasibility search over k, starting from a lower bound and increasing up to $\Delta +1$. For each fixed k, the search uses the saturation-degree rule for vertex selection and enforces the equitable size constraints that every color class must have cardinality either $\lfloor n/k\rfloor$ or $\lceil n/k\rceil$. The ILP algorithm introduces binary variables $x_{v,c}$, where $x_{v,c}=1$ indicates that vertex v is assigned color c. The formulation includes constraints requiring that each vertex receives exactly one color, adjacent vertices receive different colors, and all color classes satisfy the equitable cardinality bounds.

For each trial, we record the coloring number, the coloring assignment, the running time, and the gaps between the step-based equitable coloring and the two exact benchmark algorithms. These trial-level results are then summarized by the average coloring number and the average running time for each fixed pair $(n,m)$—see Figure 10. In terms of computational complexity, the step-based equitable coloring algorithm has a time complexity of $O\left(mn\right)$ if no fallback correction is performed. Otherwise, the fallback correction step dominates the running time, leading to a worst case complexity of $O\left(\Delta \left(G\right)n^2\right)$. In contrast, the DSATUR-based exact search is an exponential time backtracking algorithm, with worst case complexity on the order of $O\left(k^n\right)$. The ILP formulation contains $nk$ binary variables and approximately $n+\left|E\right(G\left)\right|k+2k$ main constraints, and its solution process relies on branch and bound, which is also exponential in the worst case. Therefore, the DSATUR-based and ILP algorithms are used as exact benchmarks for small scale instances, whereas the proposed step-based equitable coloring algorithm is evaluated for its ability to obtain competitive equitable coloring numbers with substantially lower computational burden.

Figure 10. Performance comparison of the three algorithms under different parameter pairs $(n,m)$. (a) Average coloring number for each fixed pair $(n,m)$. (b) Average running time for each fixed pair $(n,m)$.

Table 1 reports the comparison of coloring numbers and coloring gaps for $m=2$ and $n=10,12,\dots ,30$. Table 2 reports the comparison of running times for $m=2$ over the same range of n. Table 3 reports the average coloring numbers of the three algorithms for $n=30$ and $m=2,4,\dots ,\lfloor n/2\rfloor$. Table 4 presents the corresponding average running times of the three algorithms under the same parameter settings.

Table 1. Comparison of coloring numbers and gaps for $m=2$.

n	m	Steped k	DSATUR k	ILP k	GapS,D	GapS,I
10	2	4.4000	3.0000	3.0000	1.4000	1.4000
12	2	3.5000	2.9000	2.8000	0.6000	0.7000
14	2	4.3000	3.4000	2.9000	0.9000	1.4000
16	2	3.6000	3.2000	3.2000	0.4000	0.4000
18	2	3.7000	2.6000	2.6000	1.1000	1.1000
20	2	3.7000	2.8000	2.6000	0.9000	1.1000
22	2	4.4000	3.1000	2.9000	1.3000	1.5000
24	2	4.0000	3.3000	3.0000	0.7000	1.0000
26	2	3.8000	2.9000	2.9000	0.9000	0.9000
28	2	4.4000	3.3000	3.1000	1.1000	1.3000
30	2	4.1000	2.9000	2.7000	1.2000	1.4000

Table 2. Comparison of running times for $m=2$.

n	m	Steped Time	DSATUR Time	ILP Time
10	2	0.001072	0.001559	0.013191
12	2	0.000710	0.001279	0.007798
14	2	0.002254	0.002262	0.012252
16	2	0.000496	0.002166	0.021390
18	2	0.001560	0.001920	0.008634
20	2	0.000850	0.002749	0.013806
22	2	0.001891	0.003638	0.017729
24	2	0.000496	0.003818	0.019854
26	2	0.006463	0.022776	0.091592
28	2	0.007552	0.028871	0.062910
30	2	0.008289	0.034049	0.041481

Table 3. Comparison of coloring numbers for $n=30$.

n	m	$\mathbf{\Delta }\left(\mathit{G}\right)+1$	Steped k	DSATUR k	ILP k
30	2	5	5	3.3000	3.1000
30	4	8.7000	6.8000	4.7000	4
30	6	12.6000	8.5000	6.7000	5.5000
30	8	16.3000	15.3000	8.4000	6.5000
30	10	20.3000	20.3000	9.5000	9
30	12	24.2000	24.2000	13.3000	11.6667
30	14	28.1000	28.1000	15.7000	15

Table 4. Comparison of running times for $n=30$.

n	m	Steped Time	DSATUR Time	ILP Time
30	2	0.0028	0.0054	0.0244
30	4	0.0052	0.0334	2.2155
30	6	0.0112	0.0497	23.3775
30	8	0.0195	0.0583	36.0969
30	10	0.0303	0.0514	36.1665
30	12	0.0389	0.0672	42.2981
30	14	0.0498	0.0581	48.3496

The results show that, although the proposed step-based equitable coloring algorithm still has a gap from the optimal equitable coloring number obtained by exhaustive search, it requires much less running time. As shown in the tables and Figure 10, the proposed step-based equitable coloring algorithm already exhibits a clear advantage in running time when n is small, and this advantage can become more pronounced as n increases. Therefore, despite the gap between the equitable coloring number produced by our algorithm and the optimum, the proposed algorithm remains a meaningful and practical choice in parallel scenarios where low latency is required. For example, equitable coloring of circulant graphs is equivalent to the parallel partitioning problem in the decoding of multi-edge QC-LDPC codes for 5G communications, where fast execution and low latency are crucial.

4. Conclusions and Future Work

In this paper, the partitioning problem arising in parallel decoding of multi-edge QC-LDPC codes is reformulated as an equitable coloring problem on circulant graphs. From this perspective, upper bounds for the equitable chromatic number ${\chi }_{=}\left(G(n;D)\right)$ are derived, and several efficient equitable coloring algorithms are proposed, including pattern-based periodic coloring and step-based coloring algorithms. In some cases, the proposed methods require fewer than $\Delta \left(G\right)+1$ colors, while maintaining a computational complexity lower than $O(\Delta \left(G\right)n^2)$. The results further demonstrate that the structural symmetry of circulant graphs can be effectively exploited to obtain equitable coloring schemes. Specifically, once an equitable coloring is constructed for one symmetric unit, the corresponding coloring pattern can be periodically extended to the remaining units, thereby enabling the efficient construction of an equitable coloring for the entire graph. As a representative example, for the circulant graph $G(n;D)$, if $(d_{max}+1)\mid n$, then the equitable chromatic number satisfies ${\chi }_{=}\left(G(n;D)\right)\le d_{max}+1$, where $D=\{d_1,\dots ,d_{max}\}$ and $d_{max}$ denotes the largest element of D.

Overall, this work provides both theoretical bounds and practical algorithms for the equitable coloring of circulant graphs, thereby offering a useful framework for decoding partition design in multi-edge QC-LDPC codes. Future research will focus on applying these equitable coloring results to the search and construction of multi-edge QC-LDPC codes with high-parallel decoding partitioning capability. Moreover, further investigation of the equitable chromatic number and corresponding coloring schemes for circulant graphs is still needed, particularly with respect to tighter bounds and more general structural conditions. It is also of interest to extend the proposed ideas and techniques to broader classes of graphs, which may deepen the theoretical understanding of equitable coloring and expand its potential applications in coding theory.