4,196 research outputs found
Latin cubes of even order with forbidden entries
We consider the problem of constructing Latin cubes subject to the condition
that some symbols may not appear in certain cells. We prove that there is a
constant such that if and is a -dimensional array where every cell contains at most symbols, and
every symbol occurs at most times in every line of , then is
{\em avoidable}; that is, there is a Latin cube of order such that for
every , the symbol in position of does not
appear in the corresponding cell of .Comment: arXiv admin note: substantial text overlap with arXiv:1809.0239
Maximal partial Latin cubes
We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the numbers k for which there exists a maximal partial Latin cube of order n with exactly k entries. Finally, we prove that maximal partial Latin cubes of order n exist of each size from approximately half-full (n3/2 for even n ≥ 10 and (n3 + n)/2 for odd n ≥21) to completely full, except for when either precisely 1 or 2 cells are empty
Entanglement and quantum combinatorial designs
We introduce several classes of quantum combinatorial designs, namely quantum
Latin squares, cubes, hypercubes and a notion of orthogonality between them. A
further introduced notion, quantum orthogonal arrays, generalizes all previous
classes of designs. We show that mutually orthogonal quantum Latin arrangements
can be entangled in the same way than quantum states are entangled.
Furthermore, we show that such designs naturally define a remarkable class of
genuinely multipartite highly entangled states called -uniform, i.e.
multipartite pure states such that every reduction to parties is maximally
mixed. We derive infinitely many classes of mutually orthogonal quantum Latin
arrangements and quantum orthogonal arrays having an arbitrary large number of
columns. The corresponding multipartite -uniform states exhibit a high
persistency of entanglement, which makes them ideal candidates to develop
multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome
Ryser Type Conditions for Extending Colorings of Triples
In 1951, Ryser showed that an array whose top left subarray is filled with different symbols, each occurring at most once
in each row and at most once in each column, can be completed to a latin square
of order if and only if the number of occurrences of each symbol in is
at least . We prove a Ryser type result on extending partial coloring of
3-uniform hypergraphs. Let be finite sets with and
. When can we extend a (proper) coloring of (all triples on a ground set , each one being repeated
times) to a coloring of using the fewest
possible number of colors? It is necessary that the number of triples of each
color in is at least . Using hypergraph detachments
(Combin. Probab. Comput. 21 (2012), 483--495), we establish a necessary and
sufficient condition in terms of list coloring complete multigraphs. Using
H\"aggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295--313), we
show that the number of triples of each color being at least is
sufficient. Finally we prove an Evans type result by showing that if , then any -coloring of any subset of can be
embedded into a -coloring of as
long as .Comment: 10 page
Perfect domination in regular grid graphs
We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph of . In contrast, there is just one
1-perfect code in and one total perfect code in
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
with parallel total perfect codes, and the -perfect and
total perfect code partitions of and , the former
having as quotient graph the undirected Cayley graphs of with
generator set . For , generalization for 1-perfect codes is
provided in the integer lattice of and in the products of cycles,
with partition quotient graph taken as the undirected Cayley graph
of with generator set .Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
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