Cycle decompositions of complete multigraphs

DecomposingKu+w−Kuinto cycles of prescribed lengths

“…This result was recently generalised to complete multigraphs [6]. Partial results have also been obtained for decompositions of complete bipartite graphs [9,12,19] and complete multipartite graphs [1].…”

“…This 'switching' method was first applied to packings of the complete graph [3][4][5], and has since been generalised to other graphs [6,12,14]. See [13] for a proof of Lemma 3 and a survey of switching techniques for graph decompositions.…”

“…. , (12,6,8,8), there exists an (M, k)-packing of K U ′ ,W whose reduced leave has a decomposition into an ℓ-path and a t-path whose end vertices are in W .…”

“…3 ⌉ e if t ≥ 12 and k = 0 otherwise; (iii) 3 ≤ ℓ ≤ min(u, w) for each entry ℓ in N, and d = 0 if u = 5; (iv) if t > 0, there is some entry m in N such that m ≥ t + 2; and (v) either • a + c ≥ 6, a + 2c ≤ w, and b ≥ 1; or • a + c = 3, (m, t) ̸ = (w, 2), and (a, c, t, u, w) ̸ = (0, 3, 6,9,8). Then there exists an (N, 3 a , 4 b , 5 c , 6 d , k)-decomposition of K u+w − K u that includes cycles with lengths (3 a , 4 b , 5 c , 6 d , k) that each contain at most one pure edge.…”

“…By Lemma 23, to find such a packing it suffices to show that max(k + 2, 2a 3 + 4c 3 , 8) + t ≤ 2|U 3 | + 2 and (2a 3 + 4c 3 , t, |U 3 |, w) ̸ = (12,6,8,8) (note that 2a 3 + 4c 3 ∈ {2, 4, . .…”