{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:30:44Z","timestamp":1753882244625,"version":"3.41.2"},"reference-count":13,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01n02","funder":[{"DOI":"10.13039\/501100005270","name":"Fujian Provincial Department of Science and Technology","doi-asserted-by":"publisher","award":["2021J01860"],"award-info":[{"award-number":["2021J01860"]}],"id":[{"id":"10.13039\/501100005270","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11301217"],"award-info":[{"award-number":["11301217"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12271210"],"award-info":[{"award-number":["12271210"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"New Century Excellent Talents in Fujian Province University","award":["JA14168"],"award-info":[{"award-number":["JA14168"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Parallel Process. Lett."],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:p> 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\u00a0al. [Journal of Graph Theory\u00a04(3) (1980) 337\u2013341] 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]. <\/jats:p>","DOI":"10.1142\/s0129626423500044","type":"journal-article","created":{"date-parts":[[2023,4,22]],"date-time":"2023-04-22T05:44:00Z","timestamp":1682142240000},"source":"Crossref","is-referenced-by-count":0,"title":["4-Free Strong Digraphs with the Maximum Size"],"prefix":"10.1142","volume":"33","author":[{"given":"Qifan","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Science, Jimei University, Xiamen, Fujian 361021, PR China"},{"name":"School of Mathematics and Statistics, Fujian Normal University, Fuzhou, Fujian 350117, China"}]},{"given":"Liqiong","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Science, Jimei University, Xiamen, Fujian 361021, PR China"}]},{"given":"Yuqing","family":"Lin","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Computer Science, University of Newcastle, NSW 2308, Australia"}]}],"member":"219","published-online":{"date-parts":[[2023,4,25]]},"reference":[{"key":"S0129626423500044BIB001","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-84628-970-5","volume-title":"Graph Theory","author":"Bondy J. 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