Essential upper bounds on the total domination number

A set S of vertices in a graph G is a total dominating set of G if every vertex has a neighbor in S. The total domination number, γ_t(G), is the minimum cardinality of a total dominating set of G. Let n_G and m_G denote the number of vertices and edges, respectively, in G. We prove that all the essential upper bounds on the total domination number of a graph G without isolated vertices and isolated edges can be written in the unified form γ_t(G)≤(2an_G+2bm_G)∕(3a+2b) for constants b≥0 and a≥[Formula presented](1−b). We also provide unified forms of upper bounds for the total domination number of a graph with minimum degree δ in the cases when δ∈{2,3,4}. For example, we show that the bound γ_t(G)≤an_G+bm_G is valid for every graph G with minimum degree δ=3 if and only if both b≥0 and a≥[Formula presented](1−3b) hold.