Maximum uniformly resolvable decompositions of Kv and Kv−I

Abstract

a b s t r a c tLet K v denote the complete graph of order v and K v − I denote K v minus a 1-factor. In this article we investigate uniformly resolvable decompositions of K v and K v − I into r classes containing only copies of 3-stars and s classes containing only copies of 3-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.

Complex uniformly resolvable decompositions of K_v

“…Recently, the existence problem for H-factorizations of K v has been studied and a lot of results have been obtained, especially on the following types of uniformly resolvable H-decompositions: for a set H consisting of two complete graphs of orders at most five in [3,13,14,15]; for a set H of two or three paths on two, three, or four vertices in [5,6,9]; for H = {P 3 , K 3 + e} in [4]; for H = {K 3 , K 1,3 } in [8]; for H = {C 4 , P 3 } in [11]; for H = {K 3 , P 3 } in [12]; for 1-factors and n-stars in [7]; and for H = {P 2 , P 3 , P 4 } in [9]. In connection with our current studies the following cases are most relevant:…”

Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

“…Additional existence problems for Γ-factorizations of K v or K v -I have been studied and many results have been obtained, especially on uniformly resolvable Γ-decompositions: when Γ is a set of two complete graphs of an order of at most five in [4][5][6][7]; when Γ is a set of two or three paths on two, three or four vertices in [8][9][10]; for Γ = {P 3 , K 3 + e} in [11]; for Γ = {K 3 , K 1,3 } in [12]; for Γ = {C 4 , P 3 } in [13]; for Γ = {K 3 , P 3 } in [14]; for Γ = {K 2 , K 1,3 } in [15,16]; for Γ = {K 2 , K 1,4 } in [17]. Most famous is the variation of the Oberwolfach problem known as the Hamilton-Waterloo problem.…”

“…In this paper, we study uniformly resolvable Γ-decompositions in the case when Γ = {C n , K 1,n }. The existence problem of a (C n , K 1,n )-URD(v; r, s) was solved for n = 2 ( [9], note that C 2 = K 2 ) and n = 3 ( [12]). Here we deal with the case when n is even and greater or equal to 4.…”

The spectrum of α-resolvable λ-fold (K_4 - e)-designs

“…, ∞ 6 } is the hole. In Z 16 develop the full 2-parallel base class {(0, 3, ∞ 1 ; 12), (1, 5, ∞ 2 ; 2), (8,13, ∞ 3 ; 4), (14,15, ∞ 4 ; 11), (6, 11, ∞ 5 ; ∞ 6 ), (∞ 1 , 2, 1; 3), (∞ 2 , 4, 13; 8), (∞ 3 , 7, 0; 14), (∞ 4 , 9, 6; 10), (∞ 5 , 10, 5; 15), (∞ 6 , 12, 7; 9)}. Additionally, include the partial 2-parallel class {(0, 8, 2; 10), (1, 9, 3; 11), (2, 10, 4; 12), (3, 11, 5; 13), (4, 12, 6; 14), (5, 13, 7; 15), (6,14,8; 0), (7, 15, 9; 1)} repeated five times.…”

“…When α = 1, we simply speak of resolvable design and parallel classes. The existence problem of resolvable G-decompositions has been the subject of an extensive research (see [1,4,5,7,8,9,10,11,12,14,15,16,18,19,21,24]). The α-resolvability, with α > 1, has been studied for: G = K 3 by D. Jungnickel, R. C. Mullin, S. A. Vanstone [13], Y. Zhang and B.…”