The Geo Johan Chromatic Number and Its Boundaries for Various of Graphs

Abstract

A collection S of vertices from this set is referred to as a g(G) when every vertex present in the graph can be found lying along at least one most direct routeconnecting some pair of vertices drawn from S. The smallest possible size of such a collection is known as the geodetic number, written as g(G). Separately, a legitimate vertex colouring of a graph is described as a Johan colouring when each and every vertex in the graph possesses what is termed a rainbow neighbourhood — meaning that among all vertices adjacent to a given vertex, every colour used in the colouring appears at least once. The largest count of distinct colours that can be employed within any such valid colouring is referred to as the Johan chromatic number, denoted J(G). Building upon these two foundational ideas, this work puts forward a unified concept called the geodetic Johan chromatic set. A vertex subset qualifies as a geodetic Johan chromatic set only when it simultaneously satisfies the conditions required of both a g(G) and a J(G). The lowest cardinality achievable by any such combined set defines a new graph parameter called the Geo Johan chromatic number, represented by the notation Jgc(G). This study systematically derives the value of Jgc(G) across a variety of well-known and standard families of graphs. Beyond these specific computations, the paper also establishes tight and sharp bounds that govern this parameter for the broader family of connected graphs. Furthermore, one particularly noteworthy finding demonstrated in this work is that the geodetic Johan chromatic number does not behave monotonically with respect to the subgraph relationship — that is, moving to a subgraph does not necessarily decrease or preserve this number in a predictable direction,thesmallest cardinality of any such set is designated the Geo Johan chromatic number, expressed asJgc(G). The value ofJgc(G)is determined for various standard graph families, and sharp bounds are established for connected graphs. Furthermore, it is demonstrated that the geodetic Johan chromatic number fails to be monotone under the subgraph relation.