Abstract
Coloring of graph theory is widely used in different fields like the map coloring, traffic light problems, etc. Hypergraphs are an extension of graph theory where edges contain single or multiple vertices. This study analyzes cluster hypergraphs where cluster vertices too contain simple vertices. Coloring of cluster networks where composite/cluster vertices exist is done using the concept of coloring of cluster hypergraphs. Proper coloring and strong coloring of cluster hypergraphs have been defined. Along with these, local coloring in cluster hypergraphs is also provided. Such a cluster network, COVID19 affected network, is assumed and colored to visualize the affected regions properly.
1. 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 [1–3] 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.
There are two common ways of defining colorings of hypergraphs. The 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. This 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. They 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 -colorable if there is a proper coloring with colors such that the sizes of any two color classes differ by at most one.”
The 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. The existing coloring concepts are not adequate to this newly defined cluster hypergraph. This 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. This type of coloring is termed local coloring.
Until now, hypergraphs contain only simple nodes. This representation has problems while representing cluster/group nodes. The concept of clustering can be found as follows. Clustering algorithms [27–30] are advanced as a powerful tool to examine the massive amount of data. The 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. This 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. This issue has been rectified using fuzzy color. Thus, 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. This coloring technique has been implemented to such cases using coloring of cluster hypergraphs.
2. Hypergraphs
Definition 1. Let be a finite set and let be a family of subsets of such thatThe pair is called a hypergraph with vertex set and hyperedge set . The elements of are vertices of a hypergraph H, and the sets are hyperedges of a hypergraph H.
The length of a cycle is the number of edges in it. The girth of a hypergraph is the length of the shortest cycle it contains.
A vertex coloring of a hypergraph is called proper coloring, if any hyperedge of size greater than or equal to two contains at least two vertices of different colors.
A hypergraph consists of a finite set of vertices and a collection of subsets of . A strong coloring of is a map such that whenever for some , . The corresponding strong chromatic number is the least number of colors for which has a strong coloring.
By the degree of a vertex , denoted by , we mean the number of edges containing .
By , we denote the maximum vertex degree in the hypergraph .
3. Cluster Hypergraphs
Definition 2. (see [7]): Let be a nonempty set and be a subset of such that and . Now, be a multiset whose elements belong to such that(i).(ii)For each element , there exists at least one element such that .Then, is said to be cluster hypergraph where is said to be vertex set and is said to be multi-hyperedge set. Throughout 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 be a nonempty set and be a subset of , , such that and . Now, be a multiset whose elements belong to such that(i).(ii)For each element , there exists at least one element such that .Then, is said to be k-cluster hypergraph where is said to be vertex set and is said to be multi-hyperedge set. Generally, for , cluster hypergraphs are assumed as cluster hypergraphs.
Definition 4. The 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 is a simple as well as a maximal node, and the node is a maximal but nonsimple node.
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Definition 5. A cluster hypergraph is called a uniform cluster hypergraph if each edge of the hypergraph contains exactly nodes, and each maximal node contains simple nodes. In Figure 5, a 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.
Definition 8. A uniform cluster hypergraph is called a complete uniform cluster hypergraph if there exist an edge between any two maximal nodes and an edge between any two simple nodes within cluster nodes. An example of a complete (2, 3)-uniform cluster hypergraph is shown in Figure 8.
4. Cluster Hypergraph Coloring
A vertex coloring of a cluster hypergraph 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 consists of a finite set of vertices and a collection of subsets of V. A strong coloring of is a map such that whenever are maximal nodes and for some , .
Example 1. In Figure 4(a), a cluster hypergraph is drawn where , , and . In Figure 4(b), a proper coloring is shown. The graph has three maximal nodes . Now the edge connects all the three maximal nodes. As per the proper coloring method, two maximal nodes are to be given a different color. The 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. The graph has three maximal nodes . Now the edge connects all the three maximal nodes. Hence, as per the strong coloring method, the nodes in an edge will get separate colors. The nonmaximal nodes will get equal color as of its maximal nodes and similarly for others.
4.1. Chromatic Number
The 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). The corresponding proper chromatic number is the least number of colors for which has a proper coloring. The corresponding strong chromatic number is the least number of colors for which has a strong coloring.
Theorem 1. The 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 in a cluster hypergraph , where , are maximal nodes (see Figure 9).
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Case 1. (strong coloring): As per definition of strong coloring, each node of an edge gets separate color. Thus, size of an edge containing the maximum number of maximal nodes gets maximum color. Hence, the chromatic number based on strong coloring of a path in cluster hypergraphs is the size of an edge containing a maximum number of maximal nodes.
Case 2 (proper coloring):. As per the definition of proper coloring, two nodes of an edge get separate color. Thus, in a path, maximal nodes get alternating color. The chromatic number based on proper coloring of a path in a cluster hypergraph is two.
Theorem 2. The chromatic number based on strong coloring of a cycle in cluster hypergraphs is the size of an edge containing the maximum number of maximal nodes provided a number of such maximal nodes is greater than or equal to three; otherwise, the chromatic number is two or three.
Proof. Let us consider a cycle in a cluster hypergraph , where , are maximal nodes (see Figure 10). Also, let the maximum size of an edge in the graph be .
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Case 1. (): As per the definition of strong coloring, each node of an edge gets separate color. Thus, 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. The 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.
4.2. Bipartite or 2-Colorable Cluster Hypergraphs
A cluster hypergraph is bipartite if its maximal node set can be partitioned into two independent sets (nonadjacent node set) such that every edge connects both partite sets of maximal nodes or connects intercluster nodes. Equivalently, is a bipartite if it is 2-colorable by proper coloring.
4.3. -Partite Cluster Hypergraphs
Let us define a -partite cluster hypergraph as a -uniform cluster hypergraph whose maximal nodes can be partitioned into 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 nodes.
Example 2. In Figure 11, a 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, another uniform cluster hypergraph is colored by strong coloring technique. The chromatic number is 3.
Note 1: Let H be a cluster hypergraph of maximal vertex degree of maximal nodes, such that its each edge contains at least maximal nodes. Let . Then, the following holds.(1)The chromatic number of is by proper vertex coloring.(2)The chromatic number of is by proper vertex coloring if and .Note 2: Every minimally nonbipartite cluster hypergraph has at least as many edges as vertices.
Theorem 4. For any n-uniform cluster hypergraph , the two inequalities hold: .
Proof. The first inequality is true by observation. The second inequality can be acceptable in the following way. Each edge in any n-uniform cluster hypergraph has at most 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 of H which is less than or equal to the number of neighbors + 1. Thus, .
4.4. 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.
The local degree of a vertex f in Figure 13 is 4. The local degree of nonmaximal nodes in Figure 13 is given in Table 2.
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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 get mixed of the color BLUE and white ()%. As per Table 2, f gets the basic color BLUE, and b gets a mixture of 75% BLUE and 25% of white color. c, d, and e get a mixture of 50% BLUE and 50% white and so on (see Figure 14).
5. Scope of Applications
Cluster hypergraphs are convenient to represent any social networks. The recent literature on social network analysis can be found in [31–36]. This small example illustrates the network spread by COVID19 in worldwide. The 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. The network is shown in Figure 15. The network as a cluster hypergraph is colored by proper coloring (see Figure 16).
Benefits of the study are as follows:(1)This study analyzed cluster hypergraphs, a generalized class of hypergraphs.(2)This 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.
6. Conclusions
This 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. The centrality of a simple node is no longer important compared to the centrality of a cluster node. This study is a backbone for such applications.
Data Availability
The data in Table 3 have been collected from https://www.worldometers.info/coronavirus/ dated 05.04.2020. The data are available in the public domain.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07049321).