AbstractThe chromaticity of graphs is the term used referring to the question of chromatic equivalence and chromatic uniqueness of graphs. Since the arousal of the interest on the chromatically equivalent and chromatically unique graphs, various concepts and results under the said areas of research have been discovered and many families of such graphs have been obtained. The purpose of this thesis is to contribute new results on the chromaticity of graphs, specifically, K4 - homeomorphs with girth 9 and 6-bridge graphs. A K4 - homeomorph is a graph derived from a complete graph, K4. Such a homeomorph is denoted by K4(a, b, c, d, e, f) where the six edges of K4 are replaced by the six paths of length a, b, c, d, e and f. Let N and Ok be a set of natural numbers and a multigraph with two vertices and k edges, respectively. For any aI, a2, ... , ak E N, the graph O(al' a2, ... , ak) is a subdivision ofOk where the edged of Ok are replaced by paths of length a1,a2,...,ak,respectively. The subdivis.....