IC-Colorings and IC-Indices of graphs

Given a coloring f : V (G) → IN 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 ICcoloring 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 be M(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.