Decomposition of the complete bipartite graph with a 1-factor removed into cycles and stars

Minimum coverings of crowns with cycles and stars

“…Before plunging into the proof of the case of (n − k)(n − k − 1) ≥ k, a result due to Lee and Lin [4] is needed.…”

“…Lemma 11 [4]. If k is an even integer with k ≥ 4, then there exist k/2 − 1 edge-disjoint k-cycles in C k/2,k/2−1 ∪ K k/2,k/2 .…”

“…The existence problems for (C k , S k )-decomposition of K m,n and C n,n−1 have been completely settled by Lee [1] and Lee and Lin [4], respectively. Lee [2] obtained the maximum packing and minimum covering of the balanced complete bipartite multigraph λK n,n with (C k , S k ).…”

Decomposition of the complete bipartite multigraph into cycles and stars

“…In [26], Shyu considered the existence of a decomposition of K m,n into paths and stars with k edges, giving a necessary and sufficient condition for k = 3. Recently, Lee [16] and Lee and Lin [17] established necessary and sufficient conditions for the existence of (C k , S k )-decompositions of the complete bipartite graph and the complete bipartite graph with a 1-factor removed, respectively.…”

Decomposition of complete bipartite graphs into cycles and stars with four edges

“…The study of ðK, HÞ -multidecomposition has been introduced by Atif Abueida and M. Daven [1]. Moreover, Atif Abueida and Theresa O'Neil [2] have settled the existence of ðK, HÞ-multidecomposition of K m ðkÞ when ðK, HÞ ¼ ðK 1, nÀ1 , C n Þ for n ¼ 3, 4, 5: Priyadharsini and Muthusamy [9] established necessary and sufficient condition for the existence of ðG n , H n Þ-multidecomposition of kK n where G n , H n 2 C n , f P nÀ1 , S nÀ1 g: Lee [7], gave necessary and sufficient condition for the multidecomposition of K m,n into at least one copy of C k and S k : Lee and J.J. Lin [8], have obtained necessary and sufficient condition for the decomposition of complete bipartite graph minus a one factor into cycles and stars. Shyu [10] considered the existence of a decomposition of K m,n into paths and stars with k edges, giving a necessary and sufficient condition for k ¼ 3: Jeevadoss and Muthusamy [5] have obtained some necessary and sufficient condition for the existence of a decomposition of complete bipartite graphs into paths and cycles.…”