4-Free Strong Digraphs with the Maximum Size

Directed cycles in digraphs are useful in embedding linear arrays and rings, and are suitable for designing simple algorithm with low communication costs in parallel computer systems, thus the existence of directed cycles on digraphs has been largely investigated. Let [Formula: see text], [Formula: see text] be integers. Bermond et al. [Journal of Graph Theory 4(3) (1980) 337–341] proved that if the size of a strong digraph [Formula: see text] with order [Formula: see text] is at least [Formula: see text], then the girth of [Formula: see text] is no more than [Formula: see text]. Consequently, when [Formula: see text] is a 4-free strong digraph with order [Formula: see text], which means that every directed cycle in [Formula: see text] has length at least [Formula: see text], then the maximum size of [Formula: see text] is [Formula: see text]. In this paper, we mainly give the structural characterizations for all 4-free strong digraphs of order [Formula: see text] whose arc number exactly is [Formula: see text].