The IC-indices of complete bipartite graphs

Abstract

Let G be a connected graph, and let f be a function mapping V(G) into N. We define f(H) = Sigma(nu is an element of V(H)) f(nu) for each subgraph H of G. The function f is called an IC-coloring of G if for each integer k in the set {1, 2, ... , f(G)} there exists and (induced) connected subgraph H of G such that f(H) = k, and the IC-index of G, M(G), is the maximum value of f(G) where f is an IC-coloring of G. In this paper, we show that M(K-m,K-n) = 3.2(m+n-2)-2(m-2)+2 for each complete bipartite graph K-m,K-n, 2 <= m <= n.