The \(r\)-matching sequencibility of complete graphs

Adam Mammoliti¹

¹UNSW Sydney

The electronic journal of combinatorics, Tome 25 (2018) no. 1

Résumé

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 ${\bigl\lfloor\frac{rn-1}{2}\bigr\rfloor-1}\leq cms_r(K_n) \leq \bigl\lfloor\frac{rn-1}{2}\bigr\rfloor$ 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 $cms_r(G)$.

Détail

DOI : 10.37236/7187

Classification : 05C65, 05C70, 05C78
Mots-clés : graph, matching, edge ordering, matching sequencibility, graph decomposition, hypergraph