A counterexample to the bold conjecture

A pair of vertices (x,y) of a graph G is an o-critical pair if o(G + xy) > o(G), where G + xy denotes the graph obtained by adding the edge xy to G and o(H) is the clique number of H. The o-critical pairs are never edges in G. A maximal stable set S of G is called a forced color class of G if S meets every o-clique of G, and o-critical pairs within S form a connected graph. In 1993, G. Bacsó raised the following conjecture which implies the famous Strong Perfect Graph Conjecture: If G is a uniquely o-colorable perfect graph, then G has at least one forced color class. This conjecture is called the Bold Conjecture. Here we show a simple counterexample to it. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 165–168, 1997     

A counterexample to the rank-coloring conjecture

It has been conjectured by C. van Nuffelen that the chromatic number of any graph with at least one edge does not exceed the rank of its adjacency matrix. We give a counterexample, with chromatic number 32 and with an adjacency matrix of rank 29.     

A counterexample to Wood's conjecture

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C₀(L,C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C₀(L) being almost transitive.     

A counterexample to the triangle conjecture

The triangle conjecture sets a bound on the cardinality of a code formed by words of the form aⁱba^j. A counterexample exceeding that bound is given. This also disproves a stronger conjecture that every code is commutatively equivalent to a prefix code.     

A counterexample to Bueler's conjecture

We give a counterexample to the following conjecture of E. Bueler: ``The de Rham cohomology of any complete Riemannian manifold is isomorphic to a weighted cohomology where the weight is the heat kernel."

     

A simple counterexample to the Hambleton-Taylor-Williams conjecture

It is shown that the conjectured formula of Hambleton, Taylor, and Williams for theG-theory of the integral group ring of a finite group does not hold for the symmetric groupsS ₅.     

A counterexample to the ``composition conjecture"

In this note we construct a class of counterexamples to the ``composition conjecture" concerning an infinitesimal version of the center problem for the polynomial Abel equation in the complex domain.

     

A counterexample to the finite height conjecture

There is an infinite subdirectly irreducible lattice which generates a variety that contains only finitely many subvarietes.The author was supported in part by NSF Grant DMS 94-00511     

A counterexample to the bipartizing matching conjecture

The bipartizing matching conjecture (BMC) is a rather new approach to the nowhere zero 5-flow conjecture (NZ5FC) and the cycle double cover conjecture (CDCC). We show that the BMC is wrong in its actual version by constructing a counterexample. The construction arises from the investigation of the problem to cover the vertices of a graph by two induced Eulerian subgraphs. Finally, we state a modified version of the BMC which has the same impact on the NZ5FC and CDCC.     

A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

A counterexample to a conjecture of Friedland

This paper is concerned with a conjecture of Friedland [S. Friedland, Rational orthogonal similarity of rational symmetric matrices, Linear Algebra Appl. 192 (1993) 109–114]. A method for constructing counterexamples to the above conjecture is provided.     

A minimum dimensional counterexample to Ganea's conjecture

A 7-dimensional CW-complex having Lusternik-Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexample to Ganea's conjecture on the Lusternik-Schnirelmann category of spaces.     

A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ?^p space to a Banach space with an easily computable norm. We use that algorithm to show that Matsaev’s proposed extension of the inequality of John von Neumann is false in case p=4. Matsaev conjectured that for every contraction T on L^p (1<p<∞) one has for any polynomial P

‖P(T)‖_L^p→L^p?‖P(S)‖_{?^p(Z⁺)→?^p(Z⁺)}     

A counterexample to the Alon-Saks-Seymour conjecture and related problems

Consider a graph obtained by taking an edge disjoint union of k complete bipartite graphs. Alon, Saks, and Seymour conjectured that such graphs have chromatic number at most k+1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.