{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T08:47:12Z","timestamp":1770972432890,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Two tournaments $T_1$ and $T_2$ on the same vertex set $X$ are said to be switching equivalent if $X$ has a subset $Y$ such that $T_2$ arises from $T_1$ by switching all arcs between $Y$ and its complement $X\\setminus Y$. The main result of this paper is a characterisation of the abstract finite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral.  Moreover, if $G$ is such a group, then there is a switching class $C$, with Aut$(C)\\cong G$, such that every subgroup of $G$ of odd order is the full automorphism group of some tournament in $C$. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of random $G$-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group $G$, acting on a set $X$, is contained in the automorphism group of some switching class of tournaments with vertex set $X$ if and only if the Sylow 2-subgroups of $G$ are cyclic or dihedral and act semiregularly on $X$. Applying this result to individual permutations leads to an enumeration of switching classes, of switching classes admitting odd permutations, and of tournaments in a switching class. We conclude by remarking that both the class of switching classes of finite tournaments, and the class of \"local orders\" (that is, tournaments switching-equivalent to linear orders), give rise to countably infinite structures with interesting automorphism groups (by a theorem of Fra\u00efss\u00e9).<\/jats:p>","DOI":"10.37236\/1516","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:03:16Z","timestamp":1578708196000},"source":"Crossref","is-referenced-by-count":14,"title":["Automorphisms and Enumeration of Switching Classes of Tournaments."],"prefix":"10.37236","volume":"7","author":[{"given":"L.","family":"Babai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P. J.","family":"Cameron","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2000,8,1]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r38\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:22:36Z","timestamp":1579324956000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,8,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1516","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,8,1]]},"article-number":"R38"}}