4,196 research outputs found

    Latin cubes of even order with forbidden entries

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    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 γ>0\gamma > 0 such that if n=2tn=2t and AA is a 33-dimensional n×n×nn\times n\times n array where every cell contains at most γn\gamma n symbols, and every symbol occurs at most γn\gamma n times in every line of AA, then AA is {\em avoidable}; that is, there is a Latin cube LL of order nn such that for every 1i,j,kn1\leq i,j,k\leq n, the symbol in position (i,j,k)(i,j,k) of LL does not appear in the corresponding cell of AA.Comment: arXiv admin note: substantial text overlap with arXiv:1809.0239

    Maximal partial Latin cubes

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    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

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    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 kk-uniform, i.e. multipartite pure states such that every reduction to kk 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 kk-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

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    In 1951, Ryser showed that an n×nn\times n array LL whose top left r×sr\times s subarray is filled with nn 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 nn if and only if the number of occurrences of each symbol in LL is at least r+snr+s-n. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let X,YX,Y be finite sets with XYX\subsetneq Y and Y0(mod3)|Y|\equiv 0 \pmod 3. When can we extend a (proper) coloring of λ(X3)\lambda \binom{X}{3} (all triples on a ground set XX, each one being repeated λ\lambda times) to a coloring of λ(Y3)\lambda \binom{Y}{3} using the fewest possible number of colors? It is necessary that the number of triples of each color in (X3)\binom{X}{3} is at least X2Y/3|X|-2|Y|/3. 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 X/2Y/6|X|/2-|Y|/6 is sufficient. Finally we prove an Evans type result by showing that if Y3X|Y|\geq 3|X|, then any qq-coloring of any subset of λ(X3)\lambda \binom{X}{3} can be embedded into a λ(Y12)\lambda\binom{|Y|-1}{2}-coloring of λ(Y3)\lambda \binom{Y}{3} as long as qλ(Y12)λ(X3)/X/3q\leq \lambda \binom{|Y|-1}{2}-\lambda \binom{|X|}{3}/\lfloor{|X|/3}\rfloor.Comment: 10 page

    Perfect domination in regular grid graphs

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    We show there is an uncountable number of parallel total perfect codes in the integer lattice graph Λ{\Lambda} of R2\R^2. In contrast, there is just one 1-perfect code in Λ{\Lambda} and one total perfect code in Λ{\Lambda} restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products Cm×CnC_m\times C_n with parallel total perfect codes, and the dd-perfect and total perfect code partitions of Λ{\Lambda} and Cm×CnC_m\times C_n, the former having as quotient graph the undirected Cayley graphs of Z2d2+2d+1\Z_{2d^2+2d+1} with generator set {1,2d2}\{1,2d^2\}. For r>1r>1, generalization for 1-perfect codes is provided in the integer lattice of Rr\R^r and in the products of rr cycles, with partition quotient graph K2r+1K_{2r+1} taken as the undirected Cayley graph of Z2r+1\Z_{2r+1} with generator set {1,...,r}\{1,...,r\}.Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi

    Completion and deficiency problems

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    Given a partial Steiner triple system (STS) of order nn, 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 nn with at most rεn2r \le \varepsilon n^2 triples, it can always be embedded into a complete STS of order n+O(r)n+O(\sqrt{r}), 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 P\mathcal{P} and a graph GG, we define the deficiency of the graph GG with respect to the property P\mathcal{P} to be the smallest positive integer tt such that the join GKtG\ast K_t has property P\mathcal{P}. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a KkK_k-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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