On claw-free M-oriented critical kernel-imperfect digraphs

A kernel of a digraph D is an independent and dominating set of vertices of D. A chord of a directed cycle C = (0, 1,…,n, 0) is an arc ij of D not in C with both terminal vertices in C. A diagonal of C is a chord ij with j ≠ i − 1. Meyniel made the conjecture (now know to be false) that if D is a diagraph such that every odd directed cycle has at least two chords then D has a kernel. Here we obtain some properties of claw-free M-oriented critical kernel-imperfect digraphs. As a consequence we show that if D is an M-oriented K_1,3-free digraph such that every odd directed cycle of length at least five has two diagonals then D has a kernel. © 1996 John Wiley & Sons, Inc.     

Real 2-Regular Classes and 2-Blocks

Suppose that G is a finite group. We show that every 2-block of G has a defect class which is real. As a partial converse, we show that if G has a real 2-regular class with defect group D and if N(D)/D has no dihedral subgroup of order 8, then G has a real 2-block with defect group D. More generally, we show that every 2-block of G which is weakly regular relative to some normal subgroup N has a defect class which is real and contained in N. We give several applications of these results and also investigate some consequences of the existence of non-real 2-blocks.     

Efficient algorithms for a constrained k-tree core problem in a tree network

Let T=(V,E) be a free tree in which each vertex has a weight and each edge has a length. Let n=|V|. Given T and parameters k and l, a (k,l)-tree core is a subtree X of T with diameter l, having k leaves, which minimizes the sum of the weighted distances from all vertices in T to X. In this paper, two efficient algorithms are presented for finding a (k,l)-tree core of T. The first algorithm has O(n²) time complexity for the case that each edge has an arbitrary length. The second algorithm has O(lkn) time complexity for the case that the lengths of all edges are 1. The (k,l)-tree core problem has an application in distributed database systems.     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

A Face of a Projective Triangulation Removed for Its Geometric Realizability

Let M be a map on a surface F ². A geometric realization of M is an embedding of F ² into a Euclidian 3-space ℝ³ with no self-intersection such that each face of M is a flat polygon. In Bonnington and Nakamoto (Discrete Comput. Geom. 40:141–157, 2008), it has been proved that every triangulation G on the projective plane has a face f such that the triangulation G−f on the M?bius band obtained from G by removing the interior of f has a geometric realization. In this paper, we shall characterize such a face f of G.     

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and _R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to _R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module _R ω, we show that the projective dimension of _R ω and the right orthogonal dimension (relative to _R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that _R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to _R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to _R ω) if and only if it has a special ω^⊥-preenvelope.     

A Local Property of Large Polyhedral Maps on Compact 2-Dimensional Manifolds

We prove that each polyhedral map G on a compact 2-manifold, which has large enough vertices, contains a k-path, a path on k vertices, such that each vertex of it has, in G, degree at most 6k; this bound being best possible for k even. Moreover, if G has large enough vertices of degree >6k, than it contains a k-path such that each its vertex has degree, in G, at most 5k; this bound is best possible for any k. Received: December 8, 1997 Revised: April 27, 1998     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

On Finite Saturated Chains of Overrings

If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established.     

On the small Krull dimension of modules

We introduce and study the concept of small Krull dimension of a module which is Krull-like dimension extension of the concept of DCC on small submodules. Using this concept we extend some of the basic results for modules with this dimension, which are almost similar to the basic properties of modules with Krull dimension. When for a module A with small Krull dimension, whose Rad(A) is quotient finite dimensional, then these two dimensions for Rad(A) coincide. In particular, we prove that if an R-module A has finite hollow dimension, then A has small Krull dimension if and only if it has Krull dimension. Consequently, we show that if A has properties AB5* and qfd, then A has s.Krull dimension if and only if A has Krull dimension.     

The Rost invariant has trivial kernel for quasi-split groups of low rank

For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map . This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that R _G has trivial kernel if G is quasi-split of type E ₆ or E ₇. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank. Received: November 1, 2000     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

On finite groups whose Sylow subgroups have a bounded number of generators

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most δ elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and of index bounded by a function of δ and p. This result will be used to prove that G has a nilpotent normal subgroup of index bounded in terms of δ and p.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part II

In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value. Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

Sum and intersection of summand ideals in C(X)

Summand sum property (SSP) and summand intersection property (SIP) of modules are studied in [8] and [15] respectively. In this paper we give some topological characterizations of these properties in C(X). It is shown that the ring C(X) has SIPif and only if every intersection of closed-open subsets of Xhas a closed interior. This characterization then shows that for a large class of topological spaces, such as locally connected spaces and extremally disconnected spaces, the ring C(X) has SIP. It is also shown that C(X) has SSPif and only if the space Xhas only finitely many components. Finally, using summand ideals of C(X), we will give several algebraic characterizations of some disconnected spaces.     

Symmetric Hamilton cycle decompositions of complete graphs minus a 1‐factor

Let n≥2 be an integer. The complete graph K_n with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that K_n−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1‐factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph K_{n, n} has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of K_{n, n}, then K_{n, n}−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1‐15, 2010     

On 3-Edge-Connected Supereulerian Graphs

The supereulerian graph problem, raised by Boesch et al. (J Graph Theory 1:79–84, 1977), asks when a graph has a spanning eulerian subgraph. Pulleyblank showed that such a decision problem, even when restricted to planar graphs, is NP-complete. Jaeger and Catlin independently showed that every 4-edge-connected graph has a spanning eulerian subgraph. In 1992, Zhan showed that every 3-edge-connected, essentially 7-edge-connected graph has a spanning eulerian subgraph. It was conjectured in 1995 that every 3-edge-connected, essentially 5-edge-connected graph has a spanning eulerian subgraph. In this paper, we show that if G is a 3-edge-connected, essentially 4-edge-connected graph and if for every pair of adjacent vertices u and v, d _G (u) + d _G (v) ≥ 9, then G has a spanning eulerian subgraph.     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

Zero-Sum Flows in Regular Graphs

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that if G is an r-regular graph (r ≥ 3), then G has a zero-sum 7-flow. Furthermore, if r is divisible by 3, then G has a zero-sum 5-flow. We also show a graph of order n with a zero-sum flow has a zero-sum (n + 3)²-flow. Finally, the existence of k-flows for small graphs is investigated.