A sufficient condition for equality of arc-connectivity and minimum degree of a digraph
Let \(D\) be a simple digraph of order \(p\). For \(v \in V (D)\), \(d_ D (v)\) is the minimum of the in-degree and the out-degree of \(v\), and \(\delta (D)\) is the minimum \(d_ D (v)\), over all \(v\). The strong arc- connectivity of \(D\) is denoted by \(\lambda(D)\). It is shown that if there are \(\lfloor p/2\rfloor\) pairs of vertices \(u_ i\), \(v_ i\) of \(D\) such that \(d_ D (u_ i) + d_ D (v_ i) \geq p\), for \(i = 1, 2, \dots , \lfloor p/2 \rfloor\), then \(\lambda (D) = \delta (D)\). This generalizes a theorem of \textit{D. L. Goldsmith} and the reviewer [Discrete Math. 23, 31- 36 (1978; Zbl 0393.05035)] for simple graphs.
scientific article; zbMATH DE number 681712
scientific article