The chromatic uniqueness of complete bipartite graphs (Q1182979)

In the first part of this paper the author looks for the graph in a given class which has the maximum number of induced complete bipartite subgraphs. For graphs with a fixed number of vertices this maximum is achieved by the complete bipartite graph \(K_{n,n}\) or by \(K_{n,n+1}\). For graphs with a fixed number of edges the maximum is achieved by \(K_{1,n}\). The author determines a similar maximum for graphs with a given number of edges and a given maximum degree. In the second part of the paper the author uses a result from the first half to prove that \(K_{n,m}\) is chromatically unique when \(n,m\geq 2\), i.e., that no other graph has the same chromatic polynomial as \(K_{n,m}\).