Chromatic Equivalence And Uniqueness Of Certain 2-Connected (n, N+2)-Graphs

In the last twenty years the theory of the chromaticity of graphs has been greatly developed through the inspiration of many specialists in graph theory, motivated by theoretical and actual problems. The topic of the chromaticity of graphs is a very active one in graph theory. 'The problem of the chromaticity of a family F of graphs means the determining of the chromatically equivalent classes and the chromatically unique classes of F. Let P(G, A) denote the chromatic polynomial of a graph G. Two graphs H and G arechromatically equivalent if P(H, 2) = P(G, A.). A graph G is chromatically unique ifP(H, X) = P(G, A) implies that H isomorphic to G. The girth of a graph is the length ofthe shortest cycle.C.Y.Chao and L.C.Zhao studied the chromaticity of the family F of graphs with n+2 edges and n vertices. They first computed the chromatic polynomials of graphs in F and then divided this family into three subfamilies, F\, F2 and /^according to their chromatic polynomials and finally proved many results. K.L.Teo and K.M.Koh studied the chromaticity of the family of 2-connected (n, n+2)-graphs which contain a 4-cycle or two triangles. X.E.Chen and K.Z.Ouyang studied the chromaticity of the family of 2-connected (n, ?2)-graphs which have girth 5 and are not homeomorphic to K4.This thesis studies the chromaticity of the family W\ of 2-connected (n, n+2)-graphs which have girth 6 and are not homeomorphic to K4, and the family W2 of 2-connected (n, n+2)-graphs which have girth 7 and are not homeomorphic to K4.We divided W1 and Wi respectively into some subfamilies by exhaustion. According to the formula of Chao and Zhao, in this thesis by comparing the coefficients of chromatic polynomials of the subfamilies we determine all equivalent classes and obtain some unique graphs in W1 and Wi..