The chromatic uniqueness of certain complete t-partite graphs

Let G be a simple graph and P(G, lambda) denote the chromatic polynomial of G. Then G is said to be chromatically unique if for any simple graph H, P(H, lambda) = P(G, lambda) implies that H is isomorphic to G. A class L of graphs is called a class of chromatically normal graphs if, for any Y, G is an element of L, P(Y, lambda) = P(G, lambda) implies that Y is isomorphic to G. Let K(n(1), n(2),..., n(t)) denote a complete t-partite graph and L-1 = {K(n(1), n(2)..., n(t)) \ 0 < n(1) less than or equal to n(2) less than or equal to (...) less than or equal to n(t)}. The main results of the paper are as follows. Let G = K(n(1), n(2), ..., n(t)) is an element of L-t, t greater than or equal to 3, L(G) = {Y\ P(Y, lambda) = P(G, lambda)} and a(t) = (Sigma(1less than or equal to i < j less than or equal to t) (n(i) - n(j))(2)/ (2t))(1/2). If Sigma(i=1)(t) n(i) > ta(t)(2) + root2t(t - 1)a(t), then L(G) subset of or equal to L-t. Furthermore, if L-t is also a class of chromatically normal graphs, then G is chromatically unique. In particular, if G satisfies the condition (*) and one of the following conditions: (i) n(1) = n(2) = (...) = n(t), (ii) n(1) < n(2) < (...) < n(t), (iii) t = 3, (iv) t = 4, then G is chromatically unique. (C) 2003 Elsevier B.V. All rights reserved.