Upper Bounds for the Domination Numbers of Graphs Using Turán’s Theorem and Lovász Local Lemma

Abstract

Let G be a connected graph of order n with vertex set V(G). For positive integers a and b, a subset \(S\subseteq V(G)\) is an (a, b)-dominating set if every vertex \(v\in S\) is adjacent to at least a vertices inside S and every vertex \(v\in V{\setminus } S\) is adjacent to at least b vertices inside S. The minimum cardinality of an (a, b)-dominating set for G is called the (a, b)-domination number of G and is denoted by \(\gamma _{a,b}(G)\). There are various results on upper bounds for \(\gamma _{a,b}(G)\) when G is a regular graph or a and b are small numbers. In the first part of this paper, for a given graph G with the minimum degree of at least \(\max \{a,b\}\), we define a new graph \(G'\) associated to G and show that the independence number of this graph is related to \(\gamma _{a,b}(G)\). In the next part, using Lovasz local lemma, we give a randomized approach to improve previous results on the upper bounds for \(\gamma _{a,b}(G)\) in some special cases.