The r-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(Kn) = ⌊ n−1 2 ⌋ . 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(Kn) = ⌊ n−2 2 ⌋ . 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 > 1, and we denote the maximum such number for a graph G by msr(G) and cmsr(G) for the non-cyclic and cyclic cases, respectively. We conjecture that msr(Kn) = ⌊ rn−1 2 ⌋ and ⌊ rn−1 2 ⌋ − 1 6 cmsr(Kn) 6 ⌊ rn−1 2 ⌋ 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 Kn. We also provide bounds on msr(G) and cmsr(G) as well as results on hypergraph analogues of msr(G) and cmsr(G).