Completing the spectrum of almost resolvable cycle systems with odd cycle length

Abstract

In this paper, we construct almost resolvable cycle systems of order 4k + 1 for odd k ≥ 11. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton-Waterloo problem are also obtained.

On equitably 2‐colourable odd cycle decompositions

“…In particular, our convention is that the edge a b 0 1 will give the mixed difference b a n − (mod ), and the edge a b ĩ i will give the i-pure difference b a n ±( − ) (mod ). We point out that the constructions used here are similar to those used in [24,25] to obtain almost resolvable odd ℓ-cycle systems of order 2ℓ + 1 and 4ℓ + 1. In particular, a suitable colouring of the cycle systems built in [25] gives an equitably 2-colourable ℓ-cycle decomposition of K 2ℓ+1 when ≡ ℓ 1 (mod 4).…”

Almost resolvable k‐cycle systems with

“…In this section, we present some preliminary notation and definitions, and provide lemma for the construction of a k ‐ARCS for . We point out that similar methods have been used for many years (see ).…”

“…) presented the following open problem “The outstanding problem remains the construction of almost resolvable 2 k ‐cycle systems of order , since this will determine the spectrum for almost resolvable 2 k ‐cycle systems with the one possible exception of orders .” Since then, many authors have contributed to proving the following known conclusions. Theorem () Let . There exists a k ‐ARCS( n ) for any odd and any even , except for and except possibly for .…”

“…Theorem 1.1. ( [1,5,[9][10][11][12]14,[19][20][21]) Let ≡ 1 (mod 2 ). There exists a -ARCS( ) for any odd ≥ 3 and any even ∈ {4, 6, 8, 10, 14}, except for ( , ) ∈ {(3, 7), (3,13), (4,9)} and except possibly for ( , ) ∈ {(8, 33), (14, 57)}.…”

On the Existence of a Cyclic Near-Resolvable (6n+4)-Cycle System of 2K12n+Aldiabat1, Ibrahim2, Karim3 2019Journal of Mathematics2020View full textAdd to dashboardCite

“…The problem of constructing near-resolvable -cycle system of V has been contributed by many authors. A near-resolvable -cycle system of V has been constructed for = 4 with V ≡ 1 (mod 8) except possibly values V = 33, 41, 57 and except V = 9 (for which such a system does not exist) [21], = 10 with V ≡ 5 (mod 20) or V = 41 [22], ≥ 11 with V = 4 + 1 [23]. Recently, the existence of a near-resolvable -cycle system of 2 +1 for all ≥ 1 and ≡ 2 (mod 4) except possibly for = 2 and ≥ 14 has been proved by Wang and Cao [24].…”