Several Class Diagram Of Color Only,

Since the notion of chromatic uniqueness was first introduced in1978 by Chao and Whitehead, the search for chromatically whque graPhs has beenregarded as an important and interesting part in the studying of graph theory. TO thisday, various classes of chromdtically whque graphs have been found. In this paper,firstly we shall provide two new families of graphs Which are chromatically unique,then we shall study the strUcture and chromaticity of graphs in Whch any two colourclassses induce a tree, finaIly we apply the structural and chromatic results to discovera number of new families of chromatically unique graps.For a given simple graph G and a positive integer λ, let P(G;λ)denote thenumber of λ - colourings of the vertices of G. It is well-known that is a polynomialin λ, and we call P(G;λ) the chromatic polynomial of G. TWo graphs G and H aresaid to be chromatically equivalent, written G^H if P(G;λ) = P(H;λ). A graph G ischromatically unique if for all H^G, we have H ≌ G. A graph consisting of s pathsjoining two vertices is called an s-bridge graph. A K₄ homeomorph is obtained fromK₄ by a sequence of subdivisions of lines. Let T_r be the family of graphs G Whichpossesses an independent set partition {A₁,A_r} such that the subgrap induced byA_i■A_j in G is a tree for all i and j with l≤i≤j≤r, and T_r,l denote thefamily of graphs G belonging to T_r, the number of Whose triangles is1/3(3v(G)-2r)((r-1/2)-1.In chapter 2, we prove 5-bridge graph F(k₁,k₂,k₃, k₄,k₅), the lengths of Whosefive bridges are two distinct values, andmin(k₁,k₂,k₃, k₄,k₅)≥2 is chromaticallyumque.In chaPer 3, we prove a family of K, homeomorphs, the lengths of Whose threepaths are all equal to a, especially a is more than one, but lengths of the other threepaths are all longer than a, and not equa to with each other are chrondically whque.In chaPter 4, we determine the strUcture of the graPhs in the farnily Tr.,, and aPPlythe struhaal results to investigate the chromaticity of the graphs of the boly'In chaPer 5, we discove a nUmer of new family of chromatically whque graPhsby aPPlying the results in the chaPter 4.