On the construction of odd cycle systems

Abstract

Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n 1 k, n is odd, and 2 k divides n(n -1). We show that if these necessary conditions are sufficient for all n satisfying k I n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n = 1, 15, 21, or 25 (mod 301, and there exists a 21-cycle system of order n if and only if n = 1, 7, 15, or 21 (mod 421, n.# 7, 15.

A survey on the existence ofG‐Designs

“…The spectrum problem for C m had been settled previously for numerous values of m including several infinite families of values and all m ≤ 50 [25]. Crucial ingredients in the eventual complete solution were Sotteau's Theorem [109] and the result of Hoffman et al [76]. These results reduced the problem to one of settling the small values of n for each m. Several surveys have been written on cycle decompositions [38,42,95].…”

Decomposition of the complete graph plus a 1‐factor into cycles of equal length

“…The technique we use to prove this theorem is due to Hoffman, Lindner, and Rodger [7]. It requires several preliminary construction lemmas.…”

“…Note that for any graph G, G ffl K 0 ¼ G. The first two lemmas are due to Hoffman et al [7] and the proof is omitted.…”

Self-complementary graph decompositions

“…Perhaps the most natural question to ask about G-design is what is their spectrum, that is, for which values of n do they exist? In the case where G -C m , the spectrum remains unknown, despite having been considered for at least 25 years (see [4] for example). More recently the existence problem has been settled in the cases where G is a path [11], and where G is a star [10], and has nearly been settled when G is a graph with at most 5 vertices [1].…”