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, γt(G), is the minimum cardinality of a total dominating set of G. Let nG and mG 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 γt(G)≤(2anG+2bmG)∕(3a+2b) for constants b≥0 and a≥23(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 γt(G)≤anG+bmG is valid for every graph G with minimum degree δ=3 if and only if both b≥0 and a≥12(1−3b) hold.