  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 $cms_r(G)$.
