Embedding partial odd‐cycle systems in systems with orders in all admissible congruence classes

Abstract

At present for even integers m ≥ 4, the smallest known integer v such that any partial m-cycle system of order u can be embedded in an m-cycle system of order v is v = um/2+c(m) where c is a quadratic function of m [4][5][6]. Furthermore, it was also shown in [4,5] that any partial m-cycle system of order u can be embedded in an

Small embeddings for partial 5‐cycle systems

“…This is a significant improvement on the smallest known embeddings for partial 5‐cycle systems, v ≥10 u +5 and v ≡5 (mod 10), given by Lindner and Rodger , and v ≥10 u +7 and v ≡1, 5 (mod 10), given by Horsley and Pike . Our proof generalises the techniques used by Bryant and Horsley to prove Lindner's conjecture for partial 3‐cycle systems.…”

“…However, this result only gives embeddings of partial m ‐cycle systems of orders v ≡ m (mod 2 m ). Horsley and Pike recently showed that any partial m ‐cycle system of order u can be embedded in an m ‐cycle system of order v for all m ‐admissible v ≥ m (2 u +1)+( m −1)/2.…”

On cycle systems with specified weak chromatic number

“…We will also require Lemmas 3.2 and 3.3, which are stronger forms of results in [10]. Lemma 3.2 is only used in the proof of Lemma 3.3.…”

“…To do so, we will closely follow the method used in [10], which was in turn based heavily on methods employed in [11] and [12]. The embeddings constructed in this section are nearly as small as the smallest known embeddings for general odd cycle systems (see [12]).…”