Edge geo chromatic number of a graph

Abstract

The Edge Geodetic Number is also known as the smallest size among all sets of edges that make up Edge Geodetic sets and is represented by the symbol g1(G). Assuming G is a finite connected graph, we can define a set S contained in V(G) as an Edge Geodetic set if, for any pair of vertices u and v in set S, every edge in G lies on the shortest path between u and v. A set C contained in E(G) is referred to as a chromatic index set when it contains all of the proper k-edge colors of G. The chromatic index is determined by the set's lowest cardinality among its edges. The "Chromatic Index Number" is a term used to refer to the collection of these minimum cardinality chromatic index values and is represented by the symbol chi'(G). By combining an Edge Geodetic set with a Chromatic Index set, the Edge Geo Chromatic set of G is created, providing an innovative concept. The Edge Geo Chromatic Number, abbreviated as X, is the least cardinality among all Edge Geo Chromatic sets of G. In essence, an Edge Geodetic set and a Chromatic Index set combine to form an Edge Geo Chromatic graph. We talk about an edge geo chromatic linked graph with n orders. For a number of common graphs, we have computed the Edge Geo Chromatic Number and established its bounds. Furthermore, we have shown that when G represents a graph of diameter d, then chi'_gc(G) lessthan or equal to n - d +2.