A conjecture on triangles of graphs

We give various results related to the following conjecture: If a graphG does not contain more thank pairwise edge-disjoint triangles, then there exists a set of at most 2k edges that meets all triangles ofG. Some of the results were presented at the Conference on Combinatorics, Optimization and Statistics, Srinagar, India, August 1987. Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant 1-3-86-264.     

Rainbow triangles and the Caccetta-Häggkvist conjecture

A famous conjecture of Caccetta and Häggkvist is that in a digraph on vertices and minimum outdegree at least n/r there is a directed cycle of length or less. We consider the following generalization: in an undirected graph on vertices, any collection of disjoint sets of edges, each of size at least n/r, has a rainbow cycle of length or less. We focus on the case and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. In our main result, whenever is larger than a suitable polynomial in , we determine the maximum number of edges in an -vertex edge-colored graph where all color classes have size at most and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.     

An inequality for acute triangles

An inequality relating the tangents of half angles of a triangle that is necessary for solving Malfatti's problem is proved.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 11–14, 1992.     

Hopf's conjecture for a class of chemical kinetics equations

For quasilinear parabolic systems with a linear principal part, one investigates the question of the validity of Hopf's conjecture, concerning attractors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 57–73, 1981.     

Notes on proper class of triangles

Let T be a triangulated category and ξ a proper class of triangles.Some basics properties and diagram lemmas are proved directly from the definition of ξ.     

On a class of groups satisfying Bass' conjecture

In this paper, we use the Connes-Karoubi character liftings of the Hattori-Stallings rank in order to obtain examples of groups satisfying Bass' conjecture. We consider the nilpotence of the periodicity operator in cyclic homology and verify the conjecture for a certain class of groups which contains properly the ones studied by Eckmann. We also give an application of this method to the idempotent conjecture. Oblatum 21-III-1997 & 12-VI-1997     

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Proof of the bieberbach conjecture for a certain class of univalent functions

In the following we prove that for a given univalent function such that |a ₂| <0.867, |a _n |≦n for eachn. The method of proof is closely related to Milin’s method.     

On Ricceri’s conjecture for a class of nonlinear eigenvalue problems

In this work we give a negative answer to a conjecture proposed by B. Ricceri in the reference [B. Ricceri, A remark on a class of nonlinear eigenvalue problems, Nonlinear Anal. 69 (2008) 2964–2967] for a class of nonlinear elliptic eigenvalue problems.     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

A class number free criterion for catalan's conjecture

The Catalan conjecture asserts that the equation X^U−Y^V=1 with U,V>1 has no other solution in integers but 3²−2³=1 (Catalan's Conjecture, Academic Press, New York, 1994). We prove that, for primes U=p and V=q yielding a solution to the Catalan equation, the simultaneous conditions

     

On a conjecture of Hecke concerning elementary class number formulas

Let K be a totally real algebraic number field of class number h_K and L a totally imaginary quadratic extension of K of class number h_L. Hecke conjectured that there exists an elementary formula for the first factor h_L/h_K of the class number of L. The paper develops a theory which allows computation of h_L/h_K in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.Research supported by NSF Grant GP 20538     

A class of graphs for which the ulam conjecture holds

A median graph is a connected graph, such that for any three vertices u,ν and w there is exactly one vertex x that lies simultaneously on a shortest (u, ν)-path a shortest (ν, w)-path and a shortest (w, u)-path. It is proved that a median graph can be obtained from a one-vertex graph by an expansion procedure. From this characterization some nice properties are derived.     

A note on the degree conjecture for the Selberg class

The degree conjecture for the Selberg class of L-functions states that the degree d _F of every F ∈ is an integer. Moreover, it is expected that every F ∈ has polynomial Euler product, and that the degree ∂ _F of such an Euler product coincides with d _F . In this note we prove that a suitable continuity assumption on the degree d _F implies that ∂ _F = d _F for all F ∈ with polynomial Euler product.