完全四部图的色性

设G是一个图，P(G，λ)是G的色多项式．若P(G，λ)=P(H，λ)，则称G和H是色等价的，简单地用G～H表示．令[G]={HH～G)．若[G]={G)，称G是色唯一的．用G=K(n1，n2，n3，n4)表示完全四部图且2≤n1≤n2≤n3≤n4，得到了[G]С{K(x，y，z，w)-S｜z y w =n1 n2 n3 n4,1≤z≤y≤z≤w≤n4-1，或1≤x≤y≤z≤n3-1和w=n4U{G}，其中S是K(x，y，z，w)的某s条边组成的集合且K(x，y，z，w)-s表示从K(x，y，z，w)中删去S中所有边得到的图．从而证明了当n≥k 2，t≥2时，K(n-k，n，n，n)是色唯一的．

On the Chromaticity of Complete 4-partite Graphs

Let G be a graph and P(G,λ) the chromatic polynomial of G.Graphs G and H are said to be chromatically equivalent,simply denoted by G～H,if P(G,λ)=P(H,λ).Let [G]={H|H～G}.Graph G is said to be chromatically unique if [G]={G}.We denote by K(n1,n2,n3,n4) complete 4-partite graphs with 2(<)n1(<)n2(<)n3 n4.Let G K(n1,n2,n3,n4),we obtain that [G]{K(x,y,z,w)-S|x+y+z+w=n1+n2+n3+n4,1(<)x(<)y(<)z(<)w n4-1,or 1(<)x(<)y(<)z(<)n3-1 and w=n4}∪{G},where S is a set of some s edges of K(x,y,z,w) and K(x,y,z,w)-S denotes the graph obtained from K(x,y,z,w) by deleting all edges in S.Moreover we show that for any n(>)k+2 and k(>)2,K(n-k,n,n,n) is chromatically unique.