Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure

The Menger number of the strong product of graphs

The xy-Menger number with respect to a given integer ℓ, for every two vertices x, y in a connected graph G, denoted by ζℓ(x, y), is the maximum number of internally disjoint xy-paths whose lengths are at most ℓ in G. The Menger number of G with respect to ℓ is defined as ζℓ(G) = min{ζℓ(x, y) : x, y ∈ V(G)}. In this paper we focus on the Menger number of the strong product G1 G2 of two connected graphs G1 and G2 with at least three vertices. We show that ζℓ(G1 G2) ≥ ζℓ(G1)ζℓ(G2) and furthermore, that ζℓ+2(G1 G2) ≥ ζℓ(G1)ζℓ(G2) + ζℓ(G1) + ζℓ(G2) if both G1 and G2 have girth at least 5. These bounds are best possible, and in particular, we prove that the last inequality is reached when G1 and G2 are maximally connected graphs.Ministerio de Educación y Ciencia MTM2011-28800-C02-02Generalitat de Cataluña 1298 SGR200

Higher order rectifiability of measures via averaged discrete curvatures

We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and finite upper density $\mu$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplices spanned by the parameters.Comment: Thoroughly revised and shortened version. A bit stronger result about measures and not only sets. Cleaner statement of the main result. Concise introduction. No claims to build a general theor

Realizability of hypergraphs and Ramsey link theory

We present short simple proofs of Conway-Gordon-Sachs' theorem on graphs in 3-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on nonrealizability of certain hypergraphs (or simplicial complexes) in 4-dimensional space. The proofs use a reduction to lower dimensions which allows to exhibit relation between these results. We present a simplified exposition accessible to non-specialists in the area and to students who know basic geometry of 3-dimensional space and who are ready to learn straightforward 4-dimensional generalizations. We use elementary language (e.g. collections of points) which allows to present the main ideas without technicalities (e.g. without using the formal definition of a hypergraph).Comment: 19 pages, 11 figures; the paper is rewritten; exposition improve

The generalized 3-connectivity of Lexicographic product graphs

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $\kappa_3(G\circ H)\geq \kappa_3(G)|V(H)|$. We also give upper bounds for $\kappa_3(G\Box H)$ and $\kappa_3(G\circ H)$. Moreover, all the bounds are sharp.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1103.609