Classification Of Locally 2-Connected Compact Metric Spaces

The aim of this paper is to prove that, for compact metric spaces which do not contain infinite complete graphs, the (strong) property of being locally 2-dimensional is guaranteed just by a (weak) local connectivity condition. Specifically, we prove that a locally 2-connected, compact metric space M either contains an infinite complete graph or is surface like in the following sense: There exists a unique surface S such that S and M contain the same finite graphs. Moreover, M is embeddable in S, that is, M is homeomorphic to a subset of S.     

On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.     

A characterization of planar graphs by trémaux orders

A new characterization of planar graphs is stated in terms of an order relation on the vertices, called the Trémaux order, associated with any Trémaux spanning tree or Depth-First-Search Tree. The proof relies on the work of W. T. Tutte on the theory of crossings and the Trémaux algebraic theory of planarity developed by P. Rosenstiehl.     

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG ₁ andG ₂ (namely ifG=G ₁∪G ₂ andG ₁∩G ₂={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G ₁)+γ(G ₂)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.     

Barycentric systems and stretchability

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible.     

Universal coverings of PL-manifolds via coloured graphs

Summary By means of techniques and results concerning maps on surfaces [JS] and edge-coloured graphs representing PL-manifolds [FGG], we prove the existence of an infinite ball complexP(n), n > 1, such thatevery orientable PL-manifold of dimension n is a quotient of |P(n)| by the action of a finite index subgroup of a Fuchsian group with signature ,with h(2) = h(3) = 4 and h(n) = 2, for n > 3. The core of the proof is that all orientable PL-manifolds of dimensionn can be represented by edge-coloured graphs which are quotients of a universal graph, only depending onn.     

The asymptotic behaviour of the number of three-connected triangulations of the disk,with a reflective symmetry in a line

We consider the enumeration of the three-connected triangulations of the disk, with a reflective symmetry about a line. The asymptotic behavior is unlike that observed for rooted maps or for maps having rotational symmetry.     

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

Star clusters in independence complexes of graphs

We introduce the notion of star cluster of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engström and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic and clique numbers of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes.     

Combinatorial handles and manifolds with boundary

In this work we complete the study ofcombinatorial handles in (n+1)-coloured graphs with boundary, introduced in [G₁], [L] and [GV] for graphs with empty boundary and in [BGV] for 3-coloured graphs with boundary. In particular, we study the cancelling of a combinatorial handle from an (n+1)-coloured graph and its effects on the associated complex.Work performed under the auspices of the G.N.S.A.G.A. — C.N.R., and within the Project Geometria reale e complessa, supported by M.U.R.S.T. of Italy.     

Hom complexes and hypergraph colorings

Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.     

A Note on Locally Pathwise Connected Metric Spaces

Given any metric space, we construct its uniformly locally pathwise connected coreflection in the category of all metric spaces and uniformly continuous maps.      

Finite sheeted covering maps over 2-dimensional connected, compact Abelian groups

Let G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual . Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:

(1)

Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.

(2)

If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms.

     

关于局部紧度量空间的映象

本文借助于商映射、伪开映射和闭映射建立局部紧度量空间和几类具有某些特定性质mk-系之间的联系，作为推论.得到：双商ss-映射保持局部紧度量空间.     

On the genus of real projective spaces

We construct a crystallization of the real projective space whose associated contracted complex is minimal with respect to the number of n-simplexes. Then we compute the regular genus of , which is the minimum genus of a closed connected surface into which a crystallization of regularly embeds. Received: 7 February 2007     

Geodetic blocks of diameter three

It is shown that geodetic blocks of diameter 3 are self-centred and upper and lower geodetic critical and also lower diameter critical. Geodetic blocks of diameter 3 which are isomorphic toK _n ⁽²⁾ are characterised. The second author is on leave from the A. M. Jain College, Madras and acknowledges the financial support of the U. G. C. Teacher Fellowship for this research.