4 research outputs found
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
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
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