(Strong) Proper vertex connection of some digraphs

The (strong) proper vertex connection number spvc→(Γ), (s)pvc-number for short, is denoted as the smallest cardinality of colors required to color the digraph Γ so that Γ is (strong) properly vertex connected. The pvc-number and the spvc-number are calculated for some unique classes of digraphs in this paper, along with some fundamental results on these parameters. It is known that the pvc-number is not exceeding 3 for any strong digraph. For digraphs with pvc-number not exceeding 2, we provide some sufficient conditions. Furthermore, we prove that the spvc-number is at most 3 for any minimal strongly connected digraph, but it can be arbitrarily large for some strong digraphs. •We present some fundamental results on the proper vertex connection number and the strong proper vertex connection number of digraphs.•We determine the proper vertex connection number and the strong proper vertex connection number for some special classes of digraphs.•We give some sufficient conditions for Hamiltonian digraphs of odd order to have the proper vertex connection number at most 2.•We prove that the strong proper vertex connection number is at most 3 for any minimal strongly connected digraph, but it can be arbitrarily large for some strong digraphs.