Decomposition of K m,n into 4-cycles and 2t-cycles

Abstract

The originally published article incorrectly listed the university name as John's University.

DecomposingKu+w−Kuinto cycles of prescribed lengths

“…Theorem 19 is the main result of [7], and Theorems 20 and 21 are special cases of the main results of [12] and [9]. The other results in this section are tools that we will use in the proofs of Lemmas 28, 29 and 34.…”

“…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]. Here, we add to this body of work by addressing the question of when a complete graph with a hole admits a decomposition into cycles of arbitrary specified lengths.…”

“…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.…”

Packings and Coverings of the Complete Bipartite Graph by Octagons

“… Lemma 3.3 (see [14]) Let F be a 1-factor of complete bipartite graph K n,n . The necessary and sufficient condition for existence GD(K n, n \ F, C 8 ) is n ≡ 1 (mod 8).…”

Decomposition of Complete Bipartite Graphs into Cycles of Distinct Even Lengths

“…Chou et al [8] gave a solution when m i ∈ {4, 6, 8} for 1 ≤ i ≤ t. Chou and Fu [7] examined the case when m i ∈ {4, 2k} for 1 ≤ i ≤ t and k ≥ 3, and obtained complete solutions for k = 5 and 6. Under the same conditions on cycle lengths m i , the existence of a cycle decomposition of K n,n −I was also shown in [7] and [8], where I is a 1-factor of K n,n . The sufficiency of the necessary condition for the existence of an m-cycle system of K n,n − I was proved by Archdeacon et al [2] with possible exceptions.…”