Nonnegative Euler Characteristic Research Articles

Vertex arboricity of graphs embedded in a surface of non-negative Euler characteristic

The vertex arboricity [Formula: see text] of a graph [Formula: see text] is the minimum number of colors the vertices of the graph [Formula: see text] can be colored so that every color class induces an acyclic subgraph of [Formula: see text]. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text]. We prove that for the graph [Formula: see text] which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text] if no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, or no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, then [Formula: see text] in addition to the [Formula: see text]-regular quadrilateral mesh.

Neighbor sum distinguishing total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

A total coloring of a graph $$G$$G is a coloring of its vertices and edges such that adjacent or incident vertices and edges are not colored with the same color. A total $$[k]$$[k]-coloring of a graph $$G$$G is a total coloring of $$G$$G by using the color set $$[k]=\{1,2,\ldots ,k\}$$[k]={1,2,?,k}. Let $$f(v)$$f(v) denote the sum of the colors of a vertex $$v$$v and the colors of all incident edges of $$v$$v. A total $$[k]$$[k]-neighbor sum distinguishing-coloring of $$G$$G is a total $$[k]$$[k]-coloring of $$G$$G such that for each edge $$uv\in E(G)$$uv?E(G), $$f(u)\ne f(v)$$f(u)?f(v). Let $$G$$G be a graph which can be embedded in a surface of nonnegative Euler characteristic. In this paper, it is proved that the total neighbor sum distinguishing chromatic number of $$G$$G is $$\Delta (G)+2$$Δ(G)+2 if $$\Delta (G)\ge 14$$Δ(G)?14, where $$\Delta (G)$$Δ(G) is the maximum degree of $$G$$G.

Euler characteristics of generalized Haken manifolds

Haken n-manifolds have been defined and studied by B. Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, based upon the the theory of boundary patterns developed by K. Johannson. The Euler characteristic of a Haken manifold is analyzed and shown to be equal to the sum of the Charney-Davis invariants of the duals of the boundary complexes of the n-cells at the end of a hierarchy. These dual complexes are shown to be flag complexes. It follows that the Charney-Davis Conjecture is equivalent to the Euler Characteristic Sign Conjecture for Haken manifolds. Since the Charney-Davis invariant of a flag simplicial 3-sphere is known to be non-negative it follows that a closed Haken 4-manifold has non-negative Euler characteristic. These results hold as well for generalized Haken manifolds whose hierarchies can end with compact contractible manifolds rather than cells.

All complete intersection Calabi-Yau four-folds

We present an exhaustive, constructive, classification of the Calabi-Yau four-folds which can be described as complete intersections in products of projective spaces. A comprehensive list of 921,497 configuration matrices which represent all topologically distinct types of complete intersection Calabi-Yau four-folds is provided and can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html . The manifolds have non-negative Euler characteristics in the range 0 - 2610. This data set will be of use in a wide range of physical and mathematical applications. Nearly all of these four-folds are elliptically fibered and are thus of interest for F-theory model building.

Total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

Let G be a graph which can be embedded in a surface of nonnegative Euler characteristic. In this paper, it is proved that the total chromatic number of G is Δ(G)+1 if Δ(G) ⩾ 9, where Δ(G) is the maximum degree of G.

K-normal surfaces

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: • a minimal triangulation of a closed irreducible or a bounded hyperbolic 3-manifold contains no non-trivial k-normal sphere; • every triangulation of a closed manifold with at least 2 tetrahedra contains some non-trivial normal surface; • every manifold with boundary has only finitely many triangulations without non- trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.

An upper bound on adaptable choosability of graphs

Given a (possibly improper) edge-colouring F of a graph G , a vertex-colouring c of G is adapted to F if no colour appears at the same time on an edge and on its two endpoints. If for some integer k , a graph G is such that given any list assignment L of G , with | L ( v ) | ≥ k for all v , and any edge-colouring F of G , there exists a vertex-colouring c of G adapted to F such that c ( v ) ∈ L ( v ) for all v , then G is said to be adaptably k - choosable. The smallest k such that G is adaptably k -choosable is called the adaptable choice number and is denoted by c h a d ( G ) . This note proves that c h a d ( G ) ≤ ⌈ M a d ( G ) / 2 ⌉ + 1 , where M a d ( G ) is the maximum of 2 | E ( H ) | / | V ( H ) | over all subgraphs H of G . As a consequence, we give bounds for classes of graphs embeddable into surfaces of non-negative Euler characteristics.

Lightness of digraphs in surfaces and directed game chromatic number

The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k , and without 4-cycles, which can be embedded in a surface S . (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k ≥ 5 , these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game colouring number, and thus for the game chromatic number of (di)graphs with girth k and without 4-cycles embeddable in S . The game chromatic resp. game colouring number were introduced by Bodlaender [H.L. Bodlaender, On the complexity of some coloring games, Int. J. Found. Comput. Sci. 2 (1991) 133–147] resp. Zhu [X. Zhu, The game coloring number of planar graphs, J. Combin. Theory B 75 (1999) 245–258] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game colouring number of a directed simple forest is at most 3.

The Positive Lightness of Digraphs, Embeddable in a Surface, without 4-Cycles

Abstract The positive lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the positive lightness of digraphs with nonzero minimum in-degree and a given girth k ⩾ 5 which can be embedded in a surface S. These bounds are tight for surfaces of nonnegative Euler characteristics. We further obtain bounds for classes of graphs where 3-cycles are permitted.

Vanishing theorems and conjectures for theℓ2–homology of right-angled Coxeter groups

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex \Sigma_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of \Sigma_L is L and (2) \Sigma_L is contractible. It follows that if L is a triangulation of S^{n-1}, then \Sigma_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of \Sigma_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney-Davis Conjecture on flag triangulations of S^3. It also implies the following special case of the Hopf-Chern Conjecture: every closed 4-manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture. Conjecture: If a discrete group G acts properly on a contractible n-manifold, then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2.

On 2-Connected Spanning Subgraphs with Low Maximum Degree

Given a graph G , let a k -trestle of G be a 2-connected spanning subgraph of G of maximum degree at most k . Also, let χ ( G ) be the Euler characteristic of G . This paper shows that every 3-connected graph G has a (10−2 χ ( G ))-trestle. If χ ( G )⩽−5, this is improved to 8−2 χ ( G ), and for χ ( G )⩽−10, this is further improved to 6−2 χ ( G ). This result is shown to be best possible for almost all values of χ ( G ) by the demonstration of 3-connected graphs embedded on each surface of Euler characteristic χ ⩽0 which have no (5−2 χ )-trestle. Also, it is shown that a 4-connected graph embeddable on a surface with non-negative Euler characteristic has a 3-trestle, approaching a conjecture of Nash-Williams.

Sutured manifold hierarchies, essential laminations, and Dehn surgery

x0. Introduction A compact orientable surface F with nonnegative Euler characteristic is either a sphere, a disk, a torus, or an annulus. If a 3-manifold M contains such an essential surface, then it is said to be reducible, @-reducible, toroidal, or annular, respectively. Any such surface can be used to decompose the manifold further into simpler manifolds. We say that M is a simple manifold if it has no such surfaces. A simple manifold is expected to have a nice geometric structure. If M has nonempty boundary, then the Geometrization Theorem of Thurston for Haken manifolds says that M with boundary tori removed admits a nite volume hyperbolic structure with totally geodesic boundary. When M has no boundaries, Thurston’s Geometrization Conjecture asserts that M is either hyperbolic, or is a Seifert b er space with orbifold a sphere with at most 3 cone points. Suppose T is a torus boundary component of M. We use M( ) to denote the manifold obtained by Dehn lling on T so that the slope on T bounds a disk in the Dehn lling solid torus. When M = E(K) is the exterior of a knot K in S 3 , denote M( ) by K( ), and call it the manifold obtained by surgery on the knot K. It is well known that if M is simple then there are only nitely many Dehn llings on T which produce non simple manifolds. If M( 1) and M( 2) are non simple manifolds, then the geometric intersection number between 1 and 2, denoted by ( 1; 2), is proved to be at most 8 by Gordon [Gor1]. There are 10 dieren t cases, according to the types of nonnegative Euler characteristic surfaces in M( i). In many cases, the upper bounds for ( 1; 2) have been established, see Table 0.1.