An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edgeâ€¦ (More)

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof wasâ€¦ (More)

We prove that if G is a triangulation of the torus and Ï‡(G) 6= 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof wasâ€¦ (More)

In undergoing this life, many people always try to do and get the best. New knowledge, experience, lesson, and everything that can improve the life will be done. However, many people sometimes feelâ€¦ (More)

In order to determine the stability of directly measured pH and PCO2, and calculated [HCO3] in stored urine, 11 alkali-loaded normal subjects provided 33 spot and sixteen 24-hour mineral oil-covered,â€¦ (More)

The configuration space of n ordered disks of radius r inside the unit disk is denoted Confn,r(D). We study how the cohomology of this space depends on r. In particular, given a cohomology class ofâ€¦ (More)

We prove three theorems based on lemmas of Gromov involving the simplicial norm on stratified spaces. First, the Gromov singular fiber theorem (with proof originally sketched by Gromov) relates theâ€¦ (More)