Total domination and the Caccetta–Häggkvist conjecture

A total dominating set in a digraph $G$ is a subset $W$ of its vertices such that every vertex of $G$ has an immediate successor in $W$ . The total domination number of $G$ is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than $g\left(G\right)‐1$ , where $g\left(G\right)$ is the number of vertices of the smallest directed cycle contained in $G$ . We prove that these new conjectures are equivalent to the Caccetta–Häggkvist conjecture which asserts that $g\left(G\right)‐1<\frac{n}{r}$ in every digraph on $n$ vertices with minimum outdegree at least $r>0$ .