Matroids with many common bases

Lemos (Discrete Math. 240 (2001) 271–276) proved a conjecture of Mills (Discrete Math. 203 (1999) 195–205): for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂k. In this paper, we prove a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected.     

On Mills's conjecture on matroids with many common bases

In this paper, we shall prove a conjecture of Mills: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂k     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

A new lower bound on the number of trivially noncontractible edges in contraction critical 5-connected graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. An edge of a k-connected graph is called trivially noncontractible if its two end vertices have a common neighbor of degree k. Ando [K. Ando, Trivially noncontractible edges in a contraction critically 5-connected graph, Discrete Math. 293 (2005) 61-72] proved that a contraction critical 5-connected graph on n vertices has at least n/2 trivially noncontractible edges. Li [Xiangjun Li, Some results about the contractible edge and the domination number of graphs, Guilin, Guangxi Normal University, 2006 (in Chinese)] improved the lower bound to n+1. In this paper, the bound is improved to the statement that any contraction critical 5-connected graph on n vertices has at least trivially noncontractible edges.     

Circuits through specified edges

We prove a theorem implying the conjecture of Woodall [14] that, given any k independent edges in a (k+1)-connected graph, there is a circuit containing all of them. This implies the truth of a conjecture of Berge [1, p.214] and provides strong evidence to a conjecture of Lovász [8].     

Highly connected monochromatic subgraphs

We conjecture that for n>4(k-1) every 2-coloring of the edges of the complete graph K_n contains a k-connected monochromatic subgraph with at least n-2(k-1) vertices. This conjecture, if true, is best possible. Here we prove it for k=2, and show how to reduce it to the case n<7k-6. We prove the following result as well: for n>16k every 2-colored K_n contains a k-connected monochromatic subgraph with at least n-12k vertices.     

On a conjecture of entringer and slater

Entringer and Slater proved that every k-critically h-connected graph with h?2k+1 contains a vertex of degree h and conjectured the existence of two such vertices. We prove this conjecture.     

On n-skein isomorphisms of graphs

H. Whitney [Amer. J. Math.54 (1932), 150–168] proved that edge isomorphisms between connected graphs with at least five vertices are induced by isomorphisms and that circuit isomorphisms between 3-connected graphs are induced by isomorphisms. R. Halin and H. A. Jung [J. London Math. Soc.42 (1967), 254–256] generalized these results by showing that for n ≥ 2, n-skein isomorphisms between (n+1)-connected graphs are induced by isomorphisms. In this paper we show that for n ≥ 2, n-skein isomorphisms between 3-connected graphs having (n+1)-skeins are induced by isomorphisms.     

On 2-factors with k components

In this paper we study the minimum degree condition for a Hamiltonian graph to have a 2-factor with k components. By proving a conjecture of Faudree et al. [A note on 2-factors with two components, Discrete Math. 300 (2005) 218-224] we show the following. There exists a real number ε>0 such that for every integer k?2 there exists an integer n₀=n₀(k) such that every Hamiltonian graph G of order n?n₀ with has a 2-factor with k components.     

Spanning 3-ended trees in k-connected K 1,4-free graphs

A tree with at most m leaves is called an m-ended tree.Kyaw proved that every connected K1,4-free graph withσ4(G)n-1 contains a spanning 3-ended tree.In this paper we obtain a result for k-connected K1,4-free graphs with k 2.Let G be a k-connected K1,4-free graph of order n with k 2.Ifσk+3(G)n+2k-2,then G contains a spanning 3-ended tree.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

Special cohomology classes for modular Galois representations

Building on ideas of Vatsal [Uniform distribution of Heegner points, Invent. Math. 148(1) (2002) 1-46], Cornut [Mazur's conjecture on higher Heegner points, Invent. Math. 148(3) (2002) 495-523] proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E_/Q as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article, Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's two-dimensional ?-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.     

Kneser's conjecture,chromatic number,and homotopy

If the simplicial complex formed by the neighborhoods of points of a graph is (k ? 2)-connected then the graph is not k-colorable. As a corollary Kneser's conjecture is proved, asserting that if all n-subsets of a (2n ? k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets.     

The smallest values of algebraic connectivity for unicyclic graphs

The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let U_n be the set of all unicyclic graphs of order n. In this paper, we will provide the ordering of unicyclic graphs in U_n up to the last seven graphs according to their algebraic connectivities when n≥13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315-4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711].     

Weight Distribution of the Bases of a Matroid

Given a weighted graph, let w₁, w₂, . . . ,w_n denote the increasing sequence of all possible distinct spanning tree weights. In 1992, Mayr and Plaxton proved the following conjecture proposed by Kano: every spanning tree of weight w₁ is at most k−1 edge swaps away from some spanning tree of weight w_k. In this paper, we extend this result for matroids. We also prove that all the four conjectures due to Kano hold for matroids provided one partitions the bases of a matroid by the weight distribution of its elements instead of their weight. The author was partially supported by CNPq (Grant No. 302195/02-5) and ProNEx/CNPq (Grant No. 664107/97-4)     

Intersection theorems under dimension constraints

In Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0?t, k?n determine or estimate the maximum number of (0,1)-vectors in a k-dimensional subspace of the Euclidean n-space Rⁿ, such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the (0,1)-vectors of the same weight ω.The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used.In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21]. We also consider a diametric problem under dimension constraint.In Part II, we study the restricted case and solve the problem for t=1 and k<2ω, and also for any fixed 1?t?ω and k large.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients ak and ak+n of the product _∞(q,q)/_∞(qn,qn)=k_∑akqk have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487-501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.