Chromatic Uniqueness Of Certain 2-Connected (n,n+3)-Graphs

The graphs which we consider here are finite, undirected, simple and loopless. Let P(G, λ) denote the chromatic polynomial of a graph G . Two graphs G and H are chromatically equivalent if their polynomials are equal, i.e., P(G,λ) = P(H,λ). A graph G is chromatically unique if P(H, λ) = P(G, λ) implies that H is isomorphic to G.The notion of chromatic uniqueness was first introduced by Chao and Whitehead in 1976. Since then, various classes of chromatically unique graphs have been found successively. Let Gn, n+k be the class of 2-connected graphs with n vertices and n + k edges , which has always been an interesting topic of many researchers . Graph Gn,n is a cycle with n vertices and n edges, and R.C.Read presented the chromatic polynomial and proved the chromatic uniqueness of Gn,n. Graph Gn.n+1 is a θ -graph , and Chao and Whitehead showed that it is chromatically unique. The discussion of chromatic equivalence and chromatic uniqueness of Gn.n+2 began with Chao and Zhao, and they paved the path for their followers.In this thesis, we discuss some chromatically unique graphs of Gn.n+3. Teo and Koh have proved chromatic uniqueness of chromatic classes of 2-connected (n, n + 3) graphs with at least two triangles .Some other particular cases of Gn.n+3 have been discussed. L.C.Zhao presented that there exist 17 homeomorphic graphs in Gn.n+3, and then divided these graphs into five chromatic classes by the coefficients of the chromatic polynomials of the graphs. On the basis of L.C.Zhao's research, this thesis shows that three classes of Gn.n+3 are chromatically unique on certain conditions.