Given a bipartite graph G = (U ∪ V , E) such that | U |=| V | and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U , either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arise...

Case 1: Choose any x ∈ VI and any y ∈ ∂{x} (by hypothesis, ∂{x} has at least two elements). Let G∗ be the bipartite graph with input set V ∗ I = VI − {x}, output set V ∗ O = VO − {y}, and whose edges are the same as those of G, but with edges incident to either x or y deleted. The bipartite graph G∗ satisfies the hypothesis (1.1), because in Case 1 every proper subsetA ⊂ VI has |∂A| ≥ |A|+1, so...

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in maximal group. classify groups, and their subgroups rational points, up to conjugacy by birational map.

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

Let G be a p-mixed abelian group and R is a commutative perfect integral domain of charR = p > 0. Then, the first main result is that the group of all normalized invertible elements V (RG) is a Σ-group if and only if G is a Σ-group. In particular, the second central result is that if G is a Σ-group, the R-algebras isomorphism RA ∼= RG between the group algebras RA and RG for an arbitrary but fi...

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...

The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be reg...