On k-planar crossing numbers

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K_2k+1,q, for k?2. We prove tight bounds for complete graphs. We also study the rectilinear k-planar crossing number.     

Minimal non-1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G-e is 1-planar for every edge e of G. We prove that there are infinitely many minimal non-1-planar graphs (MN-graphs). It is known that every 6-vertex graph is 1-planar. We show that the graph K₇-K₃ is the unique 7-vertex MN-graph.     

Light subgraphs in the family of 1-planar graphs with high minimum degree

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2 ) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.     

Infinitesimal F-planar transformations

The theory of F-planar maps of Riemannian spaces and affinely connected spaces developed by J. Mike? and N. S. Sinyukov [1–6] naturally extends the theory of geodesic and holomorphic projective maps. In the present paper we find basic equations of infinitesimal F-planar maps and study these equations. The F-planar maps are maps between spaces endowed with affinor structures. The geometry of Riemannian spaces and affinely connected spaces endowed by affinor structures was investigated by A. P. Shirokov (see, e.g., [7–14]) who also studied maps between spaces of this type ([13, 14]).     

F-Planar graphs

An F-planar graph, where F is an ordered field, is a graph that can be represented in the plane F × F, with non-crossing line segments as edges. It is shown that the graph G is F-planar for some F if and only if every finite subgraph of G is planar.     

Difference Sets and Positive Exponential Sums I. General Properties

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set A the maximal size of a set whose difference set avoids A will be related to positive exponential sums using frequencies from A.     

Joins of 1-planar graphs

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once.In this paper,we study 1-planar graph joins.We prove that the join G + H is 1-planar if and only if the pair [G,H] is subgraph-majorized by one of pairs [C3 ∪ C3,C3],[C4,C4],[C4,C3],[K2,1,1,P3] in the case when both elements of the graph join have at least three vertices.If one element has at most two vertices,then we give several necessary/sufficient conditions for the bigger element.     

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.In this paper,we prove that every 1-planar graph G with maximum degree Δ(G)≥12 and girth at least five is totally(Δ(G)+1)-colorable.     

A Note on Generic Objects and Locally Finite Triangulated Categories

For a finite dimensional algebra A, we prove that the homotopy category of injective A-modules is generically trivial if and only if the derived category of all A-modules is generically trivial. Moreover we show some connections between the generic objects, locally finiteness and Krull-Gabriel dimension.     

On total colorings of some special 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we verify the total coloring conjecture for every 1-planar graph G if either Δ(G) ≥ 9 and g(G) ≥ 4, or Δ(G) ≥ 7 and g(G) ≥ 5, where Δ(G) is the maximum degree of G and g(G) is the girth of G.     

Inverses of trees

LetT be a tree with a perfect matching. It is known that in this case the adjacency matrixA ofT is invertible and thatA ^?1 is a (0, 1, ?1)-matrix. We show that in factA ^?1 is diagonally similar to a (0, 1)-matrix, hence to the adjacency matrix of a graph. We use this to provide sharp bounds on the least positive eigenvalue ofA and some general information concerning the behaviour of this eigenvalue. Some open problems raised by this work and connections with Möbius inversion on partially ordered sets are also discussed.     

The subalgebra lattice of a finite algebra

The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.     

Spectral Networks and Snakes

We apply and illustrate the techniques of spectral networks in a large collection of A _K-1 theories of class S, which we call “lifted A ₁ theories.” Our construction makes contact with Fock and Goncharov’s work on higher Teichmüller theory. In particular, we show that the Darboux coordinates on moduli spaces of flat connections which come from certain special spectral networks coincide with the Fock–Goncharov coordinates. We show, moreover, how these techniques can be used to study the BPS spectra of lifted A ₁ theories. In particular, we determine the spectrum generators for all the lifts of a simple superconformal field theory.     

On extensions of rational modules

Given a topological algebra A, we investigate when the categories of all rational A-modules and of finite-dimensional rational modules are closed under extensions inside the category of A-modules. We give a complete characterization of these two properties, in terms of a topological and a homological condition, for complete algebras. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Towards a spectral theory of graphs based on the signless Laplacian, II

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.     

Optimal 1-planar graphs which triangulate other surfaces

We show that, for any given non-spherical orientable closed surface F², there exists an optimal 1-planar graph which can be embedded on F² as a triangulation. On the other hand, we prove that there does not exist any such graph for the nonorientable closed surfaces of genus at most 3.     

Skew ring extensions and generalized monoid rings

A D-structure on a ring A with identity is a family of self-mappings indexed by the elements of a monoid G and subject to a long list of rather natural conditions. The mappings are used to define a generalization of the monoid algebra A[G]. We consider two of the simpler types of D-structure. The first is based on a homomorphism from G to End(A) and leads to a skew monoid ring. We also explore connections between these D-structures and normalizing and subnormalizing extensions. The second type of D-structure considered is built from an endomorphism of A. We use D-structures of this type to characterize rings which can be graded by a cyclic group of order 2.     

Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

Co-local subgroups of abelian groups II

In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] the notion of a co-local subgroup of an abelian group was introduced. A subgroup K of A is called co-local if the natural map is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups K of mixed, non-splitting abelian groups A with torsion subgroup t(A). We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D/t(A) is the divisible part of A/t(A)=D/t(A)⊕H/t(A), then K∩D=0, and one may assume that K⊆H. We will construct examples to show that K need not be a co-local subgroup of H. Moreover, we will investigate connections between co-local subgroups of A and A/t(A).