{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:09Z","timestamp":1753893849404,"version":"3.41.2"},"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>Alspach [ Bull. Inst. Combin. Appl., 52 (2008), pp. 7--20] defined the maximal matching sequencibility of a graph $G$, denoted $ms(G)$, to be the largest integer $s$ for which there is an ordering of the edges of $G$ such that every $s$ consecutive edges form a matching. Alspach also proved that $ms(K_n) = \\bigl\\lfloor\\frac{n-1}{2}\\bigr\\rfloor$. Brualdi et al. [Australas. J. Combin., 53 (2012), pp. 245--256] extended the definition to cyclic matching sequencibility of a graph $G$, denoted $cms(G)$, which allows cyclical orderings and proved that $cms(K_n) = \\bigl\\lfloor\\frac{n-2}{2}\\bigr\\rfloor$.In this paper, we generalise these definitions to require that every $s$ consecutive edges form a subgraph where every vertex has degree at most $r\\geq 1$, and we denote the maximum such number for a graph $G$ by $ms_r(G)$ and $cms_r(G)$ for the non-cyclic and cyclic cases, respectively. We conjecture that $ms_r(K_n) = \\bigl\\lfloor\\frac{rn-1}{2}\\bigr\\rfloor$ and ${\\bigl\\lfloor\\frac{rn-1}{2}\\bigr\\rfloor-1}\\leq cms_r(K_n)\u00a0 \\leq\u00a0 \\bigl\\lfloor\\frac{rn-1}{2}\\bigr\\rfloor$ and that both bounds are attained for some $r$ and $n$. We prove these conjectured identities for the majority of cases, by defining and characterising selected decompositions of $K_n$. We also provide bounds on $ms_r(G)$ and $cms_r(G)$ as well as results on hypergraph analogues of $ms_r(G)$ and $cms_r(G)$.<\/jats:p>","DOI":"10.37236\/7187","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:41:51Z","timestamp":1578652911000},"source":"Crossref","is-referenced-by-count":1,"title":["The $r$-Matching Sequencibility of Complete Graphs"],"prefix":"10.37236","volume":"25","author":[{"given":"Adam","family":"Mammoliti","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,1,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:41:22Z","timestamp":1579218082000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i1p6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,12]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/7187","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,1,12]]},"article-number":"P1.6"}}