On the chromaticity of complete multipartite graphs with certain edges added

Let P ( G , λ ) be the chromatic polynomial of a graph G . A graph G is chromatically unique if for any graph H , P ( H , λ ) = P ( G , λ ) implies H is isomorphic to G . For integers k ≥ 0 , t ≥ 2 , denote by K ( ( t − 1 ) × p , p + k ) the complete t -partite graph that has t − 1 partite sets of size p and one partite set of size p + k . Let K ( s , t , p , k ) be the set of graphs obtained from K ( ( t − 1 ) × p , p + k ) by adding a set S of s edges to the partite set of size p + k such that 〈 S 〉 is bipartite. If s = 1 , denote the only graph in K ( s , t , p , k ) by K + ( ( t − 1 ) × p , p + k ) . In this paper, we shall prove that for k = 0 , 1 and p + k ≥ s + 2 , each graph G ∈ K ( s , t , p , k ) is chromatically unique if and only if 〈 S 〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K + ( ( t − 1 ) × p , p + k ) is chromatically unique for k = 0 , 1 and p + k ≥ 3 .