Partitions of graphs into small and large sets

  • Authors:

  • Asen Bojilov;Yair Caro;Adriana Hansberg;Nedyalko Nenov

  • Affiliations:

  • -;-;-;-

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

Let G be a graph on n vertices. We call a subset A of the vertex set V(G)k-small if, for every vertex v@?A, deg(v)@?n-|A|+k. A subset B@?V(G) is called k-large if, for every vertex u@?B, deg(u)=|B|-k-1. Moreover, we denote by @f"k(G) the minimum integer t such that there is a partition of V(G) into tk-small sets, and by @W"k(G) the minimum integer t such that there is a partition of V(G) into tk-large sets. In this paper, we will show tight connections between k-small sets, respectively k-large sets, and the k-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both @f"k(G) and @W"k(G) and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro-Wei Theorem, Turan's Theorem and the Hansen-Zheng Theorem among other things.