The IC-indices of complete bipartite graphs (Q1010754)

Summary: Let \(G\) be a connected graph, and let \(f\) be a function mapping \(V(G)\) into \({\mathbb N}\). We define \(f(H)=\sum_{v\in{V(H)}}f(v)\) for each subgraph \(H\) of \(G\). The function \(f\) is called an IC-coloring of \(G\) if for each integer \(k\) in the set \(\{1,2,\dots,f(G)\}\) 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 \(M(K_{m,n})=3\cdot2^{m+n-2}-2^{m-2}+2\) for each complete bipartite graph \(K_{m,n},\,2\leq m\leq n\).