On a conjecture of Herzer concerning linear partitions

We prove: Let G be a finite group with a non-trivial partition (G) and a set (G) of subgroups such that any two components of (G) are contained in at least one W with W (G) and W G; then G is a p-group or a Frobenius group.From this we obtain: A finite group with a linear partition can only be an abelian p-group or a Frobenius group.Dedicated to Prof. H. Lenz on the occasion of his 70^th birthday     

On a conjecture concerning some automatic continuity theorems

Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let Φ be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism Φ^?1/Im?Φ by the fact that the radical (or strong radical) of the closure of Im?Φ has only zero as a common point with Im?Φ. This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.     

On a conjecture of Tutte concerning minimal tree numbers

A counterexample is given to a conjecture by Tutte on the minimum number of spanning trees that a 3-connected planar graph with a prescribed number of edges may have.     

On a conjecture concerning exactly covering systems of congruences

A conjecture of S. Znam [1] is disproved by means of a counterexample.     

On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K_p and K_q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4.     

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     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

On Lam's conjecture concerning signatures of quadratic forms

In this paper, we prove that formally real fields satisfying some explicit conditions on the v-invariant verify a Lam's conjecture concerning signatures of quadratic forms. In particular, it gives a short proof of a Marshall's conjecture for Pythagorean fields.     

On a conjecture of He concerning the spectral reconstruction of matrices

This paper is concerned with a recent conjecture of He (Electron. J. Comb. 14(1), 2007) on the spectral reconstruction of matrices. A counterexample is provided by using Hadamard matrices. We also give some results to the above mentioned conjecture (with slight modifications) in the positive direction.     

On Graver's conjecture concerning the rigidity problem of graphs

We show that the four-dimensional case of Graver's conjecture is not true.     

On a conjecture of Hoàng and Tu concerning perfectly orderable graphs

Hoàng and Tu [On the perfect orderability of unions of two graphs, J. Graph Theory 33 (2000) 32-43] conjectured that a weakly triangulated graph which does not contain a chordless path with six vertices is perfectly orderable. We present a counter example to this conjecture.     

On Kemnitz’ conjecture concerning lattice-points in the plane

In 1961, Erdős, Ginzburg and Ziv proved a remarkable theorem stating that each set of 2n−1 integers contains a subset of size n, the sum of whose elements is divisible by n. We will prove a similar result for pairs of integers, i.e. planar lattice-points, usually referred to as Kemnitz’ conjecture. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11B50.     

On k-domination and j-independence in graphs

Let $G$ be a graph and let $k$ and $j$ be positive integers. A subset $D$ of the vertex set of $G$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_k\left(G\right)$ is the cardinality of a smallest $k$-dominating set of $G$. A subset $I?V\left(G\right)$ is a $j$-independent set of $G$ if every vertex in $I$ has at most $j?1$ neighbors in $I$. The $j$-independence number ${\alpha }_j\left(G\right)$ is the cardinality of a largest $j$-independent set of $G$. In this work, we study the interaction between ${\gamma }_k\left(G\right)$ and ${\alpha }_j\left(G\right)$ in a graph $G$. Hereby, we generalize some known inequalities concerning these parameters and put into relation different known and new bounds on $k$-domination and $j$-independence. Finally, we will discuss several consequences that follow from the given relations, while always highlighting the symmetry that exists between these two graph invariants.     

Proof of a conjecture of Gross concerning fix-points

Research performed as a Feodor Lynen Research Fellow of the Alexander von Humboldt Foundation at Cornell University     

Short proof of a conjecture concerning split-by-nilpotent extensions

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of \(\mathop {\text {mod}}B\) inherited by \(\mathop {\text {mod}}C\). We show if B is a tilted algebra, then C is a tilted algebra.