Bounds on Total Domination in Terms of Minimum Degree

Abstract

A set S of vertices of graph G is a total dominating set, if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by "It ( G), is the minimum cardinality of a total dominating set of G. For graphs a with order n and minimum degree <5, we prove that 'Yt(G) ~ 1+1~<20 > n. Furthermore, if <5 is sufficiently large then this upper bound cannot be improved to be less than (1 + o(l)) 1+1 ~ 1l6 1+1>n. As a consequence of our main result, we verify a conjecture of Favaron et al. [4] for all graphs G with minimum at least 8.