We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in...

Let R be the ring of S-integers of an algebraic function field (in one variable) over a perfect field, where S is finite and not empty. It is shown that for every positive integer N there exist elements of R that can not be written as a sum of at most N units. Moreover, all quadratic global function fields whose rings of integers are generated by their units are determined.

Here we explore in silico an alternative to make planar eight π-electron bare ring systems with substitutions of some cyclooctatetraene ring carbon atoms by heavier group 14 elements. We found that the most stable eight membered rings with formulae C(4)Si(4)H(8), C(4)Ge(4)H(8), and C(4)Sn(4)H(8) have a perfect planar structure, enhancing delocalization energy as compared to cot.

We study steady-state configurations of intrinsically-straight elastic filaments constrained within rod-shaped bacteria that have applied forces distributed along their length. Perfect steady-state helices result from axial or azimuthal forces applied at filament ends, however azimuthal forces are required for the small pitches observed for MreB filaments within bacteria. Helix-like configurati...

The complement of a graph G is denoted by G. χ(G) denotes the chromatic number and ω(G) the clique number of G. The cycles of odd length at least five are called odd holes and the complements of odd holes are called odd anti-holes. A graph G is called perfect if, for each induced subgraph G of G, χ(G) = ω(G). Classical examples of perfect graphs consist of bipartite graphs, chordal graphs and c...

The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The rst of these three approaches yielded the rst (and to date only) proof of the SPGC; the other two remain promising to consider...

The pre-coloring extension problem consists, given a graph G and a subset of nodes to which some colors are already assigned, in finding 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....

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.