For a bipartite graph G we are able to characterize the complete intersection property of the edge subring in terms of the multiplicity and we give optimal bounds for this number. We give a method to obtain a regular sequence for the atomic ideal of G, when G is embedded on an orientable surface. We also give a graph theoretical condition for the edge subring associated with G to be Gorenstein....

We explore the geometry linking the shape of a curved mirror surface to the distortions it produces on a scene it reflects. Our analysis is local and differential. We assume a simple calibrated scene composed of lines passing through a point. We demonstrate that local information about the geometry of the surface may be recovered up to the second order from either the orientation and curvature ...

An origami (or flat structure) on a closed oriented surface, Sg, of genus g≥2 is obtained from finite collection unit Euclidean squares by gluing each right edge to left one and top bottom one. The main objects study in this note are pairs curves—filling simple curves, (α,β), Sg such that their minimal intersection equal algebraic intersection—they coherent. pair curves naturally associated wit...

Due to increasing demands on cutting tools, cutting edge preparation is of high priority because of its influence on the tool life. Current cutting edge preparation processes are mostly limited to generating simple roundings on the cutting edge. Multi-axis high precision form grinding processes offer great potential to generate defined cutting edge microgeometries. Knowledge about the relation ...

The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and NonSplitting Paths in a Tree (ENPT) [2]. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph cla...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T , and Ept(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least...

A proper non-empty subset C of the points of a linear space S = (P; L) is called line-closed if any two intersecting lines of S , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k ? 2. In addition it is shown that this cannot happen for projective spaces PG(n; q) unless q...

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have onl...

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of P, and Ept(P) is the EPT graph (i.e. the edge intersection graph) of the paths P of the tree T . These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least...