{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,7]],"date-time":"2026-06-07T08:50:59Z","timestamp":1780822259335,"version":"3.54.1"},"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>The inversion of a set $X$ of vertices in a digraph $D$ consists of reversing the direction of all arcs of $D\\langle X\\rangle$. The inversion number of an oriented graph $D$, denoted by $\\text{inv}(D)$, is the minimum number of inversions needed to transform $D$ into an acyclic oriented graph. In this paper, we study a number of problems involving the inversion number of oriented graphs. Firstly, we give bounds on $\\text{inv}(n)$, the maximum of the inversion numbers of the oriented graphs of order $n$. We show $n - O(\\sqrt{n\\log n})\u00a0 \\leq \\text{inv}(n) \\ \\leq \\ n - \\lceil \\log (n+1) \\rceil$. Secondly, we disprove a conjecture of Bang-Jensen et al. [DMCTS, 23(2), (2022)] asserting that, for every pair of oriented graphs $L$ and $R$, we have $inv(L\\Rightarrow R) =\\text{inv}(L) + \\text{inv}(R)$, where $L\\Rightarrow R$ is the oriented graph obtained from the disjoint union of $L$ and $R$ by adding all arcs from $L$ to $R$. Finally, we investigate whether, for all pairs of positive integers $k_1,k_2$, there exists an integer $f(k_1,k_2)$ such that if $D$ is an oriented graph with $\\text{inv}(D) \\geq f(k_1,k_2)$ then there is a partition $(V_1, V_2)$ of $V(D)$ such that $\\text{inv}(D\\langle V_i\\rangle) \\geq k_i$ for $i=1,2$. We show that $f(1,k)$ exists and $f(1,k)\\leq k+10$ for all positive integers $k$. Further, we show that $f(k_1,k_2)$ exists for all pairs of positive integers $k_1,k_2$ when the oriented graphs in consideration are restricted to be tournaments.<\/jats:p>","DOI":"10.37236\/12983","type":"journal-article","created":{"date-parts":[[2025,3,16]],"date-time":"2025-03-16T21:22:11Z","timestamp":1742160131000},"source":"Crossref","is-referenced-by-count":5,"title":["Problems, Proofs, and Disproofs on the Inversion Number"],"prefix":"10.37236","volume":"32","author":[{"given":"Guillaume","family":"Aubian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fr\u00e9d\u00e9ric","family":"Havet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Florian","family":"H\u00f6rsch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Felix","family":"Kingelhoefer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Nisse","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cl\u00e9ment","family":"Rambaud","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Quentin","family":"Vermande","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2025,3,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p42\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,16]],"date-time":"2025-03-16T21:22:11Z","timestamp":1742160131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i1p42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1,17]]}},"URL":"https:\/\/doi.org\/10.37236\/12983","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,14]]},"article-number":"P1.42"}}