A Study Of IC-coloring Of Graphs

Let G=(V, E) be a graph and let/:Vâ†'N be a function.For any subgraph H of G,define fs(H)=Σv∈v(H) f(v).In particular,we denote fs(H) by S(f). The function is called an IC-coloring if for any integer k€[1, fs(G)]there is a connected subgraph H of 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-coloing of G}.Since the IC-coloring problem has been proposed, Penrice, E.Salehi, SinMin Lee, J.F.Fink, and others have researched it, and have concluded the IC-index of complete graphs and star graphs. However, they only gave the upper bounds and lower bounds about the IC-index of path, cycle, wheels and so on.So in this article we mainly aimed improving the lower bounds of path, cycle and wheels, studied on the IC-coloring of cartesian product graphs and some special join graphs, then we could obtain results about their upper bounds and lower bounds of IC-index.