Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Designing Bioactive Delivery Systems for Tissue Regeneration

The direct infusion of macromolecules into defect sites generally does not impart adequate physiological responses. Without the protection of delivery systems, inductive molecules may likely redistribute away from their desired locale and are vulnerable to degradation. In order to achieve efficacy, large doses supplied at interval time periods are necessary, often at great expense and ensuing detrimental side effects. The selection of a delivery system plays an important role in the rate of re-growth and functionality of regenerating tissue: not only do the release kinetics of inductive molecules and their consequent bioactivities need to be considered, but also how the delivery system interacts and integrates with its surrounding host environment. In the current review, we describe the means of release of macromolecules from hydrogels, polymeric microspheres, and porous scaffolds along with the selection and utilization of bioactive delivery systems in a variety of tissue-engineering strategies

Backbone colorings for graphs

We study backbone colorings, a variation on classical vertex colorings: Given a graph G=(V,E) and a spanning subgraph H (the backbone) of G, a backbone coloring for G and H is a proper vertex coloring V ¿{ 1,2,... } in which the colors assigned to adjacent vertices in H differ by at least two. We concentrate on the cases where the backbone is either a spanning tree or a spanning path. For tree backbones of G, the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number ¿(G); for path backbones this factor is roughly 3 2 . In the special case of split graphs G, the difference from ¿(G) is at most an additive constant 2 or 1, for tree backbones and path backbones, respectively. The computational complexity of the problem ‘Given a graph G, a spanning tree T of G, and an integer l, is there a backbone coloring for G and T with at most l colors?’ jumps from polynomial to NP-complete between l¿=¿4 (easy for all spanning trees) and l¿=¿5 (difficult even for spanning paths)

Adhesion molecule expression in local-macrophage-depleted rats bearing orthotopic corneal allografts

Background: Rejection of corneal grafts is dependent on influx of T lymphocytes and macrophages. This process is partly regulated by adhesion molecules. Earlier investigations showed that corneal graft rejection in rats could be prevented by clodronate liposomes that selectively eliminate macrophages. In the present study the effect of macrophage depletion on adhesion molecule expression after corneal allotransplantation was investigated. Methods: Orthotopic corneal allografts were performed, after which rats received subconjunctival. injections with clodronate liposomes or remained untreated. On various postoperative days, grafted rats were killed and mid-eye sections were stained for expression of ICAM-1 (CD54) and beta(2)-integrins (CD18 and CD11b/c). Results: In the clodronate liposome-treated group grafts were not rejected, while in untreated rats grafts had a mean survival time of 12 days. During the first postoperative days a slightly enhanced expression of. ICAM-1 in the conjunctiva and allografted cornea of clodronate liposome-treated recipients was seen. On day 12, however, ICAM-1 expression was markedly downregulated in the allografts of this treated group. The expression of beta(2)-integrins was also significantly decreased in the allografts and recipient corneas of treated rats at this time point. Conclusion: Prolonged corneal graft survival in rats, obtained via local depletion of macrophages, correlates with diminished expression of adhesion molecules