Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars

Abstract

Let and denote a path and a star with edges, respectively. For graphs , , and , a -multidecomposition of is a partition of the edge set of into copies of and copies of with at least one copy of and at least one copy of . In this paper, necessary and sufficient conditions for the existence of the (, )-multidecomposition of the balanced complete bipartite graph are given.

Minimum coverings of crowns with cycles and stars

“…Trivially, |E(C (q−1)k+1,(q−1)k )|, |E(K (q−1)k,k+r )| and |E(K k+r,(q−1)k )| are multiples of k, by Lemmas 1 and 2, we have that C (q−1)k+1,(q−1)k , K (q−1)k,k+r and K k+r,(q−1)k have S k -decompositions A (1) , A (2) and A (3) with A (1) = (q − 1)((q − 1)k + 1), A (2) = A (3) = (k + r)(q − 1). For the case of r = 0, by Lemma 4, C k+1,k has a C k -decomposition C with |C | = k + 1.…”

“…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 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.…”

Decompositions of balanced complete bipartite graphs into suns and stars

“…Abueida and O'Neil [3] settled the existence problem for {C k , S k−1 }-decomposition of the complete multigraph λK n for k ∈ {3, 4, 5}. Recently, Lee [12,13] established necessary and sufficient conditions for the existence of a {C k , S k }-decomposition of a complete bipartite graph and {P k , S k }-decomposition of a balanced complete bipartite graph. In this paper, we consider the existence of {S(C k/2 ), S k }-decompositions of the balanced complete bipartite graph, giving necessary and sufficient conditions.…”