MINIMAL CONNECTED GEO CHROMATIC NUMBER OF SOME STANDARD GRAPHS

Abstract

For a connected graph G of order n ≥ 2, a connected geo chromatic set S???????? in a connected graph G is called a minimal connected geo chromatic set if no proper subset of S???????? is a connected geo chromatic set of G. The minimal connected geo chromatic number χ???????? + (G) is the maximum cardinality of a minimum connected geo chromatic set of G. We determined the minimum connected geo chromatic number of certain standard graphs and bounds of the minimum connected geo chromatic number is proved. It is shown that for positive integers x, y and z such that 2 ≤ x < y ≤ z, there exists a connected graph G such that ????(G) = x, χ????????(G) = y and χ???????? + (G)= z .