More on positive commutators

Let A and B be positive operators on a Banach lattice E such that the commutator C=AB−BA is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bra?i? et al. (in press) [3]. If the sum A+B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC−CA is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.     

Polynomial algorithm for sharp upper bound of rainbow connection number of maximal outerplanar graphs

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2011) [5] proved that computing rc(G) is NP-hard and deciding if rc(G)=2 is NP-complete. When edges of G are colored with fixed number k of colors, Kratochvil [6] proposed a question: what is the complexity of deciding whether G is rainbow connected? is this an FPT problem? In this paper, we prove that any maximal outerplanar graph is k rainbow connected for suitably large k and can be given a rainbow coloring in polynomial time.     

Cellularity in polyhedra

A compact subset X of a polyhedron P is cellular in P if there is a pseudoisotropy of P shrinking precisely X to a point. A proper surjection between polyhedra f:P→Q is cellular if each point inverse of f is cellular in P. It is shown that if f:P→Q is a cellular map and either P or Q is a generalized n-manifold, n≠4, then f is approximable by homeomorphisms. Also, if P or Q is an n-manifold with boundary, n≠4, 5, then a cellular map f:P→Q is approximable by homeomorphisms. A cellularity criterion for a special class of cell-like sets in polyhedra is established.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

On the genus and thickness of graphs

The genus γ(G) of a simple graph G is the minimum genus of the orientable surface on which G is embeddable. The thickness θ(G) of G is the minimum number of planar subgraphs of G whose union is G. From the definitions, it is clear that θ(G) = 1 if and only if γ(G) = 0. In this paper, we will show that θ(G) ≦ γ(G) + 1, if G has no triangle or if G is toroidal.     

Edge-Ramsey theory

LetK be a configuration, a set of points in some finite-dimensional Euclidean space. Letn andk be positive integers. The notationR(K, n, r) is an abbreviation for the following statement: For everyr-coloring of the points of then-dimensional Euclidean space,R ⁿ , a monochromatic configurationL which is congruent toK exists. A configurationK is Ramsey if the following holds: For every positive integerr, a positive integern=n(K, r) exists such that, for allm≥n, R(K, m, r) holds. A configuration is spherical if it can be embedded in the surface of a sphere inn-space, providedn is sufficiently large. It is relatively easy to show that if a configuration is Ramsey, it must be spherical. Accordingly, a good fraction of the research efforts in Euclidean Ramsey theory is devoted to determining which spherical configurations are Ramsey. It is known that then-dimensional measure polytopes (the higher-dimensional analogs of a cube), then-dimensional simplex, and the regular polyhedra inR ² andR ³ are Ramsey. Now letE denote a set of edges in a configurationK. The pair (K, E) is called an edge-configuration, andR _e (K, E, n, r) is used as an abbreviation for the following statement: For anyr-coloring of the edges ofR ⁿ , there is an edge configuration (L, F) congruent to (K, E) so that all edges inF are assigned the same color. An edge-configuration isedge-Ramsey if, for allr≥1, a positive integern=n(K, E, r) exists so that ifm≥n, the statementR _e (K, E, m, r) holds. IfK is a regular polytope, it is saidK isedge-Ramsey when the configuration determined by the set of edges of minimum length is edge-Ramsey. It is known that then-dimensional simplex is edge-Ramsey and that the nodes of any edge-Ramsey configuration can be partitioned into two spherical sets. Furthermore, the edges of any edge-Ramsey configuration must all have the same length. It is conjectured that the unit square is edge-Ramsey, and it is natural to ask the more general question: Which regular polytopes are edge-Ramsey? In this article it is shown that then-dimensional measure polytope and then-dimensional cross polytope are edge-Ramsey. It is also shown that these two infinite families and then-dimensional simplexes are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.     

The influence of weakly-supplemented subgroups on the structure of finite groups

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the paper it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of P∩G′ is weakly-supplemented in N _G (P), where P is a Sylow p-subgroup of G. As applications, some interesting results with weakly-supplemented minimal subgroups of P∩G′ are obtained.     

The Sarkovskii order for periodic continua

Suppose f is a map of a continuum X onto itself. A periodic continuum of f is a subcontinuum K of X such that fn[K]=K for some positive integer n. A proper periodic continuum of f is a periodic continuum of f that is a proper subcontinuum of X. A proper periodic continuum of f is maximal if and only if X is the only periodic continuum that properly contains it. In this paper it is shown that the maximal proper periodic continua of a map of a hereditarily decomposable chainable continuum onto itself follow the Sarkovskii order, provided the maximal proper periodic continua are disjoint. The case in which the Sarkovskii order does not hold reduces to the scenario in which the map's domain is the union of two overlapping period-two continua, each of which is maximal.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

Retractions of split graphs and End-orthodox split graphs

A retraction f of a graph G is an edge-preserving mapping of G with f(v)=v for all v∈V(H), where H is the subgraph induced by the range of f. A graph G is called End-orthodox (End-regular) if its endomorphism monoid End X is orthodox (regular) in the semigroup sense. It is known that a graph is End-orthodox if it is End-regular and the composition of any two retractions is also a retraction. The retractions of split graphs are given and End-orthodox split graphs are characterized.     

Regular separable graphs of minimum order with given diameter

A (d, c, v)-graph G is one which is regular of degree v and has diameter d and connectivity c. G is said to be minimum if it is of minimum order, i.e. has the minimumnumber of points; G is separable if c=1.In this paper, the minimum order of a (d, 1, v)-graph is determined and the construction of all minimum (d, 1, v)-graphs is described.     

Sequential closure in product spaces

Let X denote the product of m-many second countable Hausdorff spaces. Main theorems: (1) If S?X is invariant under compositions, m is weakly accessible (resp., nonmeasurable), and F?S is sequentially closed and a sequential G_σ-set which is invariant under projections for finite sets (resp., F?S is sequentially open and sequentially closed), then F is closed. (2) If S?X is invariant under projections and m is nonmeasurable, then every sequentially continuous {0, 1} valued function on S is continuous. (3) A sequentially continuous {0, 1}-valued function on an m-adic space of nonmeasurable weight is continuous. Now let X denote the product of arbitrarily many W-spaces and S?X be invariant under compositions. (4) Then in S, the closure of any Q-open subset coincides with its sequential closure.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

The Erd?s-Ko-Rado properties of set systems defined by double partitions

Let F be a family of subsets of a finite set V. The star ofFatv∈V is the sub-family {A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r} by F^(r).A double partitionP of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P)induced byP is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting sub-family of F(P)^(r) is larger than the largest star of F(P)^(r). We also characterise the cases when every extremal intersecting sub-family of F(P)^(r) is a star of F(P)^(r).     

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k?i-intersecting family. Furthermore, we discuss and Erd?s-Ko-Rado-type question forL(A), as well.     

Partitioning graphs into complete and empty graphs

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j+k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j,k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j,k)-graph. The split-chromatic number of G is the minimum j where G is a (j,j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j,k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

Semidirect product division algebras

SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC _p C _q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.