Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

Abstract

A k‐cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k‐cycles, the edges of which partition the edges of G. A k‐cycle system of λKv is known as a λ‐fold k‐cycle system of order V. A k‐cycle system of λKv (V, C) is said to be enclosed in a k‐cycle system of (λ+m)Kv+u (V∪U,P) if C⊂P and u,m≥1. We settle the difficult enclosing problem for λ‐fold 5‐cycle systems with u = 1.

Enclosings of decompositions of complete multigraphs in 2‐factorizations

“…Let  ′ be the resulting partial path decomposition in which each color class contains at least max{0, − } edges. If (iii) holds, that is, = 3 and = 1 and > , then  ′ is the required path decomposition since 2 = 3 is the number of edges in 3 . Otherwise, since ≥ ( − 1) − 1 whenever (i) or (ii) hold, we can repeatedly apply Lemma 4.1 starting from  ′ to obtain a path decomposition of of size in which each color class contains at least max{0, − } edges.…”

“…We remark that, in the situation where the multiplicities of the two graphs are not equal, past work has focused on decompositions in which each subgraph of the target decomposition is isomorphic to a cycle of one fixed length [1][2][3][4]8,13,14,17,18] or where each subgraph of the target decomposition is isomorphic to a cycle of one of a number of fixed lengths [12].…”

Extended near Skolem sequences Part I

“…Asplund et al [2] used extended Skolem sequences to construct enclosings (for information on these see [10]) of ‐fold 5‐cycle systems.…”

Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

“…There are a number of enclosing results in the case where λ < µ and the target decomposition consists of color classes each isomorphic to a cycle of one prescribed length [1,3,4,9,14,15,19,20] or each isomorphic to a cycle of one of a number of prescribed lengths [2,13]. However, less is known in the case where λ < µ and the target decomposition consists of spanning subgraphs.…”