The rr-matching sequencibility of complete graphs

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 $cms_r(G)$

The rr-matching sequencibility of complete multi-kk-partite kk-graphs

Alspach [{\sl Bull. Inst. Combin. Appl.}~{\bf 52} (2008), 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. In this paper, we consider the natural analogue for hypergraphs of this and related results and determine $ms(\lambda\mathcal{K}_{n_1,\ldots, n_k})$ where $\lambda\mathcal{K}_{n_1,\ldots, n_k}$ denotes the multi-$k$-partite $k$-graph with edge multiplicity $\lambda$ and parts of sizes $n_1,\ldots,n_k$, respectively. It turns out that these invariants may be given surprisingly precise and somewhat elegant descriptions, in a much more general setting

The cyclic matching sequenceability of regular graphs

The cyclic matching sequenceability of a simple graph $G$, denoted $\mathrm{cms}(G)$, is the largest integer $s$ for which there exists a cyclic ordering of the edges of $G$ so that every set of $s$ consecutive edges forms a matching. In this paper we consider the minimum cyclic matching sequenceability of $k$-regular graphs. We completely determine this for $2$-regular graphs, and give bounds for $k \geq 3$.Comment: 24 pages, 1 figur