Concepts on Coloring of Cluster Hypergraphs with Application

Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk, WB 721636, India Division of Applied Mathematics, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Republic of Korea School of Computer Science, University of Technology Sydney, Ultimo, Australia School of Engineering, RMIT University, Carlton, Victoria 3053, Australia Department of Mathematics, D. J. H. School, Dantan, WB 721451, India


Introduction
A hypergraph is a generalization of a graph in which any subset of a vertex set is an edge rather than two vertex sets. Specially, Berge [1][2][3] introduced hypergraphs as generalization of graph theory. Burosch and Cecherini [4] characterized cube-hypergraphs, where each hyperedge contains three vertices. Sonntag and Teichert [5,6] defined hypertrees, and they extended the notion to competition hypergraphs in another paper in 2004. To capture the notion of a cluster node, cluster hypergraphs have been introduced by Samanta et al. [7]. Uniformity and completeness properties of cluster hypergraphs have been developed here.
ere are two common ways of defining colorings of hypergraphs. e first was introduced by Erdős and Hajnal [8,9] and involves that no edge is monochromatic. Such a coloring is sometimes referred to as weak coloring. In the other case, strong coloring/rainbow coloring is done, in such a way that no two vertices belonging to a single edge share the same color [10]. Alon and Bregman [11] introduced 2 coloring in regular and uniform hypergraphs. is work has been extended by Vishwanathan [12]. Frieze and Mubayi [13] introduced the coloring in simple hypergraphs. Chang and Lawler [14] discussed edge coloring of hypergraphs. ey also solved the conjecture of Erdos, Faber, and Lovász. Furmanczyk and Obszarski [15] extended the coloring concepts to the equitable coloring of a hypergraph. Beck [16] discussed some properties on 3-chromatic hypergraphs. Obszarski and Jastrzębski [17] added more results on edge coloring of 3-uniform hypergraphs to their earlier studies [15]. Nešetřil et al. [18] defined achromatic number of simple hypergraphs. In hypergraphs, a polychromatic coloring [19] is a coloring of its vertices such that every hyperedge comprises at least one vertex of each color. A polychromatic m-coloring of a hypergraph resembles a cover m-decomposition of its dual hypergraph. Král'a et al. [20] proved that feasible sets of mixed hypertrees are gap-free. Equitable colorings of a uniform hypergraph [21] deal with an extremal problem.
Dvořák and Postle [22] presented the idea of so-called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was analyzed in detail by Bernshteyn, Kostochka, and Pron for graphs and multigraphs. DP-degree [23] for hypergraphs has been extended, with reminiscence that "a vertex coloring of a hypergraph is called proper if there are no monochromatic edges under this coloring. A hypergraph is said to be equitably r-colorable if there is a proper coloring with r colors such that the sizes of any two color classes differ by at most one." e chronological literature review is shown in Table 1 and Figure 1. All the mentioned studies focused on the vertex and edge coloring of hypergraphs. Here, the cluster node concepts have been considered for coloring of nodes. e existing coloring concepts are not adequate to this newly defined cluster hypergraph. is study implements the two ways of coloring, proper and strong coloring in cluster hypergraphs. Additionally, the coloring technique of nodes inside the cluster has been depicted. is type of coloring is termed local coloring.
Until now, hypergraphs contain only simple nodes. is representation has problems while representing cluster/ group nodes. e concept of clustering can be found as follows. Clustering algorithms [27][28][29][30] are advanced as a powerful tool to examine the massive amount of data. e focal goal of these algorithms is to categorize the data in clusters of objects, so that data in each cluster are similar based on precise criteria and data from two dissimilar clusters be different as much as possible. is study initiated the concept of cluster hypergraphs [7] with several properties. Coloring on hypergraphs is an old phenomenon. As this study presents about coloring of cluster nodes, simple nodes inside the cluster nodes get the same color. is issue has been rectified using fuzzy color. us, this paper introduces an entirely new idea on coloring, i.e., fuzzy coloring on crisp cluster hypergraphs. At present, representation of COVID19 affected regions is focused by researchers. is coloring technique has been implemented to such cases using coloring of cluster hypergraphs.

Hypergraphs
Definition 1. Let U � u 1 , u 2 , . . . , u n be a finite set and let E � e 1 , e 2 , . . . , e m be a family of subsets of U such that e pair (U, E) is called a hypergraph with vertex set U and hyperedge set E. e elements u 1 , u 2 , . . . , u n of U are vertices of a hypergraph H, and the sets e 1 , e 2 , . . . , e m are hyperedges of a hypergraph H. e length of a cycle is the number of edges in it. e girth of a hypergraph is the length of the shortest cycle it contains.
A vertex coloring of a hypergraph H is called proper coloring, if any hyperedge of size greater than or equal to two contains at least two vertices of different colors. e corresponding strong chromatic number χ s (H) is the least number of colors for which G has a strong coloring.
By the degree of a vertex v, denoted by deg v, we mean the number of edges containing v.
By Δ(H), we denote the maximum vertex degree in the hypergraph H.

Cluster Hypergraphs
Definition 2. (see [7]): Let X be a nonempty set and V X be a subset of P(X) such that ϕ ∉ V X and X ⊂ V X . Now, E be a multiset whose elements belong to P(P(X)) such that en, G � (V X , E) is said to be cluster hypergraph where V X is said to be vertex set and E is said to be multi-hyperedge set. roughout this paper, hyperedges are termed edges. An example of cluster hypergraph is shown in Figure 2, and its corresponding virtual representation is shown in Figure 3.
Definition 3. Let X be a nonempty set and V X be a subset of P k (X), k � 1, 2, 3 . . ., such that ϕ ∉ V X and X ⊂ V X . Now, E be a multiset whose elements belong to P(V X ) such that en, G � (V X , E) is said to be k-cluster hypergraph where V X is said to be vertex set and E is said to be multihyperedge set. Generally, for k � 1, 1−cluster hypergraphs are assumed as cluster hypergraphs.

Definition 4.
e nodes which are not contained in any other cluster nodes are called maximal nodes. A simple node may be termed a maximal node if it does not belong to any other nodes. For example, in Figure 4(a), the node g is a simple as well as a maximal node, and the node a, b { } is a maximal but nonsimple node.

Definition 5.
A cluster hypergraph is called a (m, n)−uniform cluster hypergraph if each edge of the hypergraph contains exactly m nodes, and each maximal node contains n simple nodes. In Figure 5, a (2, 3)−uniform cluster hypergraph is shown.

Definition 6.
A cluster hypergraph is called a cluster connected cluster hypergraph (CCCH) if there are edges which connect only maximal cluster nodes. In Figure 6, a CCCH has been drawn.

Definition 7.
A CCCH is called a complete cluster hypergraph if there exists an edge between any two maximal nodes. An example of a complete CCCH is shown in Figure 7.      Mathematical Problems in Engineering

Cluster Hypergraph Coloring
A vertex coloring of a cluster hypergraph H is called proper coloring, if any edge connecting at least two maximal nodes contains at least two such maximal nodes of different colors. All the nonmaximal nodes are given the same color as the corresponding maximal nodes.
A cluster hypergraph H � (V, E) consists of a finite set V � V (H) of vertices and a collection E � E(H) ⊆ P(V) of subsets of V. A strong coloring of G is a map Ψ : V(H) ⟶ N such that whenever u, v are maximal nodes and u, v ∈ e for some e ∈ E(G), Ψ(u) ≠ Ψ(v). Figure 4(a), a cluster hypergraph Figure 4(b), a proper coloring is shown. e graph has three maximal nodes

Example 1. In
{ }, g connects all the three maximal nodes. As per the proper coloring method, two maximal nodes are to be given a different color. e nonmaximal nodes will get the same color as of its maximal nodes and similarly for others. In Figure 4(c), strong coloring has been shown. e graph has three maximal nodes c � a, b {   edge a, b { }, d, e { }, g connects all the three maximal nodes. Hence, as per the strong coloring method, the nodes in an edge will get separate colors. e nonmaximal nodes will get equal color as of its maximal nodes and similarly for others.

Chromatic Number.
e minimum number of color to use the proper coloring (or strong coloring) method for coloring cluster hypergraphs is termed chromatic number based on proper coloring (or strong coloring). e corresponding proper chromatic number χ(H) is the least number of colors for which H has a proper coloring. e corresponding strong chromatic number χ s (H) is the least number of colors for which H has a strong coloring.

Theorem 1. e chromatic number based on strong coloring (or proper coloring) of a path in a cluster hypergraph is size of an edge containing the maximum number of maximal nodes (or two).
Proof. Let us consider a path v 1 , e 1 , v 2 , e 2 , . . . , v m in a cluster hypergraph H � (V X , E), where v i , i � 1, 2, . . . m, are maximal nodes (see Figure 9).   Figure 10). Also, let the maximum size of an edge in the graph be k.
Case 1 (k ≥ 3): As per the definition of strong coloring, each node of an edge gets separate color. us, the size of an edge containing the maximum number of maximal nodes gets maximum color. Hence, the chromatic number based on strong coloring of a cycle in a cluster hypergraph is size of an edge containing the maximum number of maximal nodes.
Case 2 (k � 2): As the cluster hypergraph contains edges of size two, for odd length cycles, the chromatic number is three, and for even length cycles, the chromatic number is two.

Corollary 1.
e chromatic number based on proper coloring of a cycle in cluster hypergraphs is two provided the maximum size of an edge is greater than or equal to three. If the maximum size of an edge is equal to two, even length cycles have chromatic number two and three for odd length cycles.

Theorem 3. Chromatic number of a CCCH is at least two and at most equal to the number of maximal nodes of the graph.
Proof. Let us consider a CCCH. Now the edges may contain two maximal nodes at least. In that case, the chromatic number is the number of maximal nodes in the graph. If the edges contain more than two vertices, then the chromatic number may reduce to two, if all nodes are included in a single edge.

n-Partite Cluster Hypergraphs.
Let us define a n-partite cluster hypergraph as a n-uniform cluster hypergraph whose maximal nodes can be partitioned into n independent sets (nonadjacent node sets) and each edge is incident to exactly one maximal node from each partition or is incident to intramaximal cluster nodes only. It is to be mentioned that if there is an edge in a maximal cluster, the edge may contain any number of nodes, not necessarily n nodes.
Example 2. In Figure 11, a (3, 3)−uniform cluster hypergraph has been shown. It can be noted that edges connect three maximal nodes of three independent sets. Also, edges may connect intercluster nodes. Naturally, the chromatic number is 2 for this case. In Figure 12

Proof.
e first inequality is true by observation. e second inequality can be acceptable in the following way. Each edge in any n-uniform cluster hypergraph has at most nΔ(H) − n neighbors of maximal nodes. Hence, for a certain edge e even if all the adjacent edges have various colors, the algorithm has always generalized to chromatic number χ(H) of H which is less than or equal to the number of neighbors + 1.

Local Coloring on Nonmaximal Nodes.
In proper coloring or strong coloring, maximal nodes are given colors. But simple or cluster nodes inside maximal nodes are given the same color as of their corresponding maximal nodes. To differ the color of nonmaximal nodes, local degree is assumed. Local degree of a nonmaximal node is the number of intramaximal node edges incident to the node. e local degree of a vertex f in Figure 13 is 4. e local degree of nonmaximal nodes in Figure 13 is given in Table 2.
Based on the normalized value of local degree, fuzzy colors are given to the nonmaximal nodes. If a maximal node gets a color say "BLUE", then the nodes within the maximal node with normalized value w get mixed of the color BLUE 100w% and white (100 − 100w)%. As per Table 2, f gets the  Figure 14).

Scope of Applications
Cluster hypergraphs are convenient to represent any social networks. e recent literature on social network analysis can be found in [31][32][33][34][35][36]. is small example illustrates the network spread by COVID19 in worldwide. e taken data are shown in Table 3. It can be noted that nearby countries (based on a certain distance) form a cluster. If one cluster is affected by other clusters, then one edge is used to connect the clusters. e network is shown in Figure 15. e network as a cluster hypergraph is colored by proper coloring (see Figure 16).
Benefits of the study are as follows: (1) is study analyzed cluster hypergraphs, a generalized class of hypergraphs.
(2) is study provided several properties of cluster hypergraphs, including a CCCH, a complete CCCH, and a uniform CCCH. (3) Proper coloring and strong coloring have been defined for cluster hypergraphs. Local coloring for nonmaximal nodes has also been introduced. (4) Proper coloring of COVID19 affected regions has been shown.

Conclusions
is study developed basic terminologies of cluster hypergraphs. Coloring of cluster hypergraphs are analyzed by two basic techniques. Proper coloring and strong coloring are two ways of coloring of cluster hypergraphs. Additionally, the coloring of intranodes of a maximal node is depicted. Finally, a small network is considered for the coloring purpose.
In future, several complex network problems can be solved by cluster hypergraph coloring/labelling. In particular, every social network is a cluster hypergraph. e centrality of a simple node is no longer important compared to the centrality of a cluster node. is study is a backbone for such applications.
Data Availability e data in Table 3 have been collected from https://www. worldometers.info/coronavirus/ dated 05.04.2020. e data are available in the public domain.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 9