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.

Let H denote a finite simple hypergraph. The cover ideal of H, denoted by J = J(H), is the monomial ideal whose minimal generators correspond to the minimal vertex covers of H. We give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J . Furthermore, we study the sets Ass(R/Js) by exploring th...

A new way of constructing (minimal) perfect hash functions is described. The technique considerably reduces the overhead associated with resolving buckets in two-level hashing schemes. Evaluating a hash function requires just one multiplication and a few additions apart from primitive bit operations. The number of accesses to memory is two, one of which is to a fixed location. This improves the...

We consider the problem of covering the complete r-uniform hypergraphs on n vertices using complete r-partite graphs. We obtain lower bounds on the size of such a covering. For small values of r our result implies a lower bound of Ω( e r r √ r n log n) on the size of any such covering. This improves the previous bound of Ω(rn log n) due to Snir [5]. We also obtain good lower bounds on the size ...

A linear (qd, q, t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a set φ1, . . . , φs of linear functionals from V to F with the following property: for all t subsets X ⊆ V there exists i ∈ {1, . . . , s} such that φi is injective when restricted to F . A linear (qd, q, t)-perfect hash family of minimal size d(t− 1) is said to be optimal. I...

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. Recently, Gronau et al determined the existence of super-simple (v, 5, 2)-BIBDs with possible exceptions of v ∈ {75, 95, 115, 135, 195, 215, 231, 285, 365, 385, 515}. In this article, we investigate the existence of a super-simple (v, 5, 4)-BIBD and show that such a design exi...