The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

The pre-coloring extension problem consists, given a graph G and a subset of nodes to which some colors are already assigned, in nding a coloring of G with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. W...

We obtain the characterization of saturation class and approximation spaces for perfect spline approximation. Also, a generalized perfect spline approximation is investigated.

A method to generate a large number of M-ary Perfect Periodic Autocorrelation (PPAC) codes of length N, with good correlation properties, is presented. N+1 perfect sequences are obtained if an inverse discrete Fourier transform (IDFT) is applied to a specific subset of N+1 bipolar sequences derived from maximal-length sequences. This set of perfect sequences has a maximum absolute value of peri...

We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation 12 = 1 + 2 + 3 + 6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpiński. We discuss some rules for generating near-perfe...

A 1-perfect code Cn q is called Hamiltonian if its minimum distance graph G(Cn q ) contains a Hamiltonian cycle. In this paper, for all admissible lengths n ≥ 13, we construct Hamiltonian nonlinear ternary 1-perfect codes, and for all admissible lengths n ≥ 21, we construct Hamiltonian nonlinear quaternary 1-perfect codes. The existence of Hamiltonian nonlinear q-ary 1-perfect codes of length N...

This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n > 1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton’s theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n)1+o(1).

In this work, we investigate in the problem of perfect matchings with prescribed matchings in the n-dimensional hypercube network Qn. We obtain the following contributions: For any arbitrary matching with at most n − 1 edges, it can be extended to a perfect matching of Qn for n ≥ 1. Furthermore, for any arbitrary non-forbidden matching with n edges, it also can be extended to a perfect matching...

The purpose of this paper is to study the information ratio of perfect secret sharing of product of some special families of graphs. We seek to prove that the information ratio of prism graphs $Y_{n}$ are equal to $frac{7}{4}$ for any $ngeq 5$, and we will gave a partial answer to a question of Csirmaz cite{CL}. We will also study the information ratio of two other families $C_{m}times C_{n}$ a...