IC-Colorings and IC-Indices of graphs

Given a coloring f:V(G)→N of graph G and any subgraph H⊂G we define fs(H)=∑v∈V(H)f(v). In particular, we denote fs(G) by S(f). The coloring f is called an IC-coloring if for any integer k∈[1,S(f)] there is a connected subgraph H⊂G such that fs(H)=k. Also, we define the IC-index of G to beM(G)=max{S(f):f is an IC-coloring of G}.In this paper we examine some well-known classes of graphs and determine their IC-indices. In addition, several conjectures are proposed.