Classification of complete 5-partite graphs and chromaticity of 5-partite graphs with 5n vertices

For a graph G,P(G,a)denotes the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent,denoted by G~H, if P(G,λ) =p(H, λ). Let [G]= {H|H ~G}. If [G]= {G},then G is said tobe chromatically unique. For a complete 5-partite graph G with 5n vertices, define θ(G) = (α(G, 6) - 2^n+l - 2^n-1 + 5)/2^n-2,where α(G, 6) denotes the number of 6-independent partitions of G. In this paper, the authors show that θ(G)≥0 and determine all graphs with θ (G) = 0, 1, 2, 5/2, 7/2, 4, 17/4. By using these resuhs the chromaticity of 5-partite graphs of the form G-S with θ(G) = 0, 1,2,5/2,7/2,4, 17/4 is investigated,where S is a set of edges of G. Many new chromatically unique 5-partite graphs are obtained.