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.
Highlights
A hypergraph is a generalization of a graph in which any subset of a vertex set is an edge rather than two vertex sets
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
Representation of COVID19 affected regions is focused by researchers. is coloring technique has been implemented to such cases using coloring of cluster hypergraphs
Summary
A hypergraph is a generalization of a graph in which any subset of a vertex set is an edge rather than two vertex sets. Furmanczyk and Obszarski [15] extended the coloring concepts to the equitable coloring of a hypergraph. Obszarski and Jastrzebski [17] added more results on edge coloring of 3-uniform hypergraphs to their earlier studies [15]. All the mentioned studies focused on the vertex and edge coloring of hypergraphs. 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. Is study initiated the concept of cluster hypergraphs [7] with several properties. Us, this paper introduces an entirely new idea on coloring, i.e., fuzzy coloring on crisp cluster hypergraphs. Is coloring technique has been implemented to such cases using coloring of cluster hypergraphs Representation of COVID19 affected regions is focused by researchers. is coloring technique has been implemented to such cases using coloring of cluster hypergraphs
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