254 research outputs found
The Brown-Colbourn conjecture on zeros of reliability polynomials is false
We give counterexamples to the Brown-Colbourn conjecture on reliability
polynomials, in both its univariate and multivariate forms. The multivariate
Brown-Colbourn conjecture is false already for the complete graph K_4. The
univariate Brown-Colbourn conjecture is false for certain simple planar graphs
obtained from K_4 by parallel and series expansion of edges. We show, in fact,
that a graph has the multivariate Brown-Colbourn property if and only if it is
series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the
exposition. To appear in Journal of Combinatorial Theory
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
Structure of Cubic Lehman Matrices
A pair of square -matrices is called a \emph{Lehman pair} if
for some integer . In this case and
are called \emph{Lehman matrices}. This terminology arises because Lehman
showed that the rows with the fewest ones in any non-degenerate minimally
nonideal (mni) matrix form a square Lehman submatrix of . Lehman
matrices with are essentially equivalent to \emph{partitionable graphs}
(also known as -graphs), so have been heavily studied as part
of attempts to directly classify minimal imperfect graphs. In this paper, we
view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite
graph, focusing in particular on the case where the graph is cubic. From this
perspective, we identify two constructions that generate cubic Lehman graphs
from smaller Lehman graphs. The most prolific of these constructions involves
repeatedly replacing suitable pairs of edges with a particular -vertex
subgraph that we call a -rung ladder segment. Two decades ago, L\"{u}tolf \&
Margot initiated a computational study of mni matrices and constructed a
catalogue containing (among other things) a listing of all cubic Lehman
matrices with of order up to . We verify their catalogue
(which has just one omission), and extend the computational results to matrices. Of the cubic Lehman matrices (with ) of order
up to , only two do not arise from our -rung ladder
construction. However these exceptions can be derived from our second
construction, and so our two constructions cover all known cubic Lehman
matrices with
Excluding Kuratowski graphs and their duals from binary matroids
We consider some applications of our characterisation of the internally
4-connected binary matroids with no M(K3,3)-minor. We characterise the
internally 4-connected binary matroids with no minor in some subset of
{M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We
also describe a practical algorithm for testing whether a binary matroid has a
minor in the subset. In addition we characterise the growth-rate of binary
matroids with no M(K3,3)-minor, and we show that a binary matroid with no
M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change
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