The IC-Indices of Complete Bipartite Graphs - Shiue, Chin-Lin; Fu, Hung-Lin (The Electronic Journal of Combinatorics, 2008)
Let G be a connected graph, and let f be a function mapping V(G) into N. We define f(H)=_vV(H) f(v) 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 an (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,n)=3·2^m+n-2-2^m-2+2 for each complete bipartite graph K_m,n, 2≤ m≤ n.