A c-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of size c, in which every possible v-tuple of elements occurs exactly once. In the binary case, i.e. where c = 2, these perfect factors have previously been studied by Etzion, [2], who showed that the obvious necessary conditions for their existence are in fact sufficient. This result has recently been ...

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.

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...