Several Three Chart-color Only

For the purpose of tackling the four-color problem,Birknoff in 1912[2]introduced the chromatic polynomial of a map M,denoted by P(M,λ),which is the number of properλ-colorings of a map M.If one could prove that P(M,4)>0 for all maps M, then this would give a positive answer to the four-color problem. For a graph G,a mapφ:Vâ†'{1,2,…,k} is called a vertex coloring of G.A coloring is proper if no two adjacent vertices hav the same color.A graph G is k-colorable if G has a proper k-coloring.The chromatic number of G,denoted byχ(G),is the minimum k such that G is k-colorable.The number of proper colorings of G with at mostλcolors is called the chromatic polynomial.denoted by P(G,λ).For a positive integer r,a partition {A1,A2,…,Ar} of V(G)is called an r-independent partition of a graph G if every Ai is an nonempty independent set of G.Letα(G,r)denote the number of r-independent partition V(G). Then'the chromatic polynomial of G is written as: where (λ)i=λ(λ-1)(λ-2)…(λ-i+1).Two graphs G and H are said to be chromatically equivalent,denoted by G～H, if P(G,λ)=P(H,λ).By [G] we denote the equivalence class determined by G under "～".In 1978,Chao and Whitehead [6] define a graph to be chromatically unique if no other graphs share its chromatic polynomial.In this thesis,the chromaticity of some graphs G=K(n,n,n)-S are chiefly studied by means of the properties of chromatic polynomials and adjoint polynomials. The author shows that for some G=K(n,n,n)-S with n≥3 and 1≤s≤n-2,if the number of 4-independent partitions of G is smaller or bigger,then G isχ-unique.