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 Lovász local lemma, we give a randomized approach to improve previous results on the upper bounds for $\gamma _{a,b}(G)$ in some special cases.