Abstract
A potential function fG of a finite, simple and undirected graph G=(V,E) is an arbitrary function fG:V(G)âN0 that assigns a nonnegative integer to every vertex of a graph G. In this paper we define the iterative process of computing the step potential functionqG such that qG(v)â¤dG(v) for all vâV(G). We use this function in the development of new CaroâWei-type and Brooks-type bounds for the independence number Îą(G) and the Grundy number Î(G). In particular, we prove that Î(G)â¤Q(G)+1, where Q(G)=max{qG(v)|vâV(G)} and Îą(G)âĽâvâV(G)(qG(v)+1)â1. This also establishes new bounds for the number of colors used by the algorithm Greedy and the size of an independent set generated by a suitably modified version of the classical algorithm GreedyMAX.
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