Abstract:Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph λK n×m can be decomposed into cycles of lengths c 1 m, . . . , c k m (plus a 1-factor if the degree is odd) whenever there exists a decomposition of λmK n into cycles of lengths c 1 , . . . , c k (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph λK n×m .
Let
true(
0X
h
true) be the collection of all
h‐subsets of an
n‐set
X
⊇
Y. Given a coloring (partition) of a set
S
⊆
true(
0.0ptX
h
true), we are interested in finding conditions under which this coloring is extendible to a coloring of
true(
0X
h
true) so that the number of times each element of
X appears in each color class (all sets of the same color) is the same number
r. The case
S
=
∅
,
r
=
1 was studied by Sylvester in the 18th century and remained open until the 1970s. The case
h
=
2
,
r
=
1 is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For
S
=
true(
0.0ptY
h
true), we settle the cases
h
=
4
,
∣
X
∣
≥
4.847323
∣
Y
∣, and
h
=
5
,
∣
X
∣
≥
6.285214
∣
Y
∣ completely. Moreover, we make partial progress toward solving the case where
S
=
true(
0.0ptX
h
true)
\
true(
0.0ptY
h
true). These results can be seen as extensions of the famous Baranyai’s theorem, and make progress toward settling a 40‐year‐old problem posed by Cameron.
Can we color the n 3 cells of an n × n × n cube L with n 2 colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that L ij = L j i = L ij for distinct i, j, ∈ {1, . . . , n}, and L iij = L jji , L iji = L jij , L ijj = L jii for i, j ∈ {1, . . . , n}? Using transportation networks, we show that such a coloring is possible if and only if n ≡ 0, 2 (mod 3) (with two exceptions, n = 1 and n = 3). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940's. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.
A cycle of length t in a hypergraph is an alternating sequence v1,e1,v2⋯,vt,et of distinct vertices vi and distinct edges ei so that false{vi,vi+1false}⊆ei (with vt+1:=v1). Let λKnh be the λ‐fold n‐vertex complete h‐graph. Let G=(V,E) be a hypergraph all of whose edges are of size at least h, and 2≤c1≤⋯≤ck≤false|Vfalse|. In order to partition the edge set of scriptG into cycles of specified lengths c1,⋯,ck, an obvious necessary condition is that 0true∑i=1kci=|E|. We show that this condition is sufficient in the following cases.
(R1)h≥max{ck,false⌈n/2false⌉+1}.
(R2)G=λKnh, h≥⌈n/2⌉+2.
(R3)G=Knh, c1=⋯=ck:=c, c|n(n−1),n≥85.In (R2), we guarantee that each cycle is almost regular. In (R3), we also solve the case where a “small” subset L of edges of Knh is removed.
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