371 research outputs found
On two conjectures of Maurer concerning basis graphs of matroids
We characterize 2-dimensional complexes associated canonically with basis
graphs of matroids as simply connected triangle-square complexes satisfying
some local conditions. This proves a version of a (disproved) conjecture by
Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14
(1973), 216-240). We also establish Conjecture 1 from the same paper about the
redundancy of the conditions in the characterization of basis graphs. We
indicate positive-curvature-like aspects of the local properties of the studied
complexes. We characterize similarly the corresponding 2-dimensional complexes
of even -matroids.Comment: 28 page
Note on Hamiltonicity of basis graphs of even delta-matroids
We show that the basis graph of an even delta-matroid is Hamiltonian if it
has more than two vertices. More strongly, we prove that for two distinct edges
and sharing a common end, it has a Hamiltonian cycle using and
avoiding unless it has at most two vertices or it is a cycle of length at
most four. We also prove that if the basis graph is not a hypercube graph, then
each vertex belongs to cycles of every length , and each edge
belongs to cycles of every length . For the last theorem, we
provide two proofs, one of which uses the result of Naddef (1984) on polytopes
and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the
other is a direct proof using various properties of even delta-matroids. Our
theorems generalize the analogous results for matroids by Holzmann and Harary
(1972) and Bondy and Ingleton (1976).Comment: 10 pages, 2 figures. Corrected a typ
Interlace Polynomials for Multimatroids and Delta-Matroids
We provide a unified framework in which the interlace polynomial and several
related graph polynomials are defined more generally for multimatroids and
delta-matroids. Using combinatorial properties of multimatroids rather than
graph-theoretical arguments, we find that various known results about these
polynomials, including their recursive relations, are both more efficiently and
more generally obtained. In addition, we obtain several interrelationships and
results for polynomials on multimatroids and delta-matroids that correspond to
new interrelationships and results for the corresponding graphs polynomials. As
a tool we prove the equivalence of tight 3-matroids and delta-matroids closed
under the operations of twist and loop complementation, called vf-safe
delta-matroids. This result is of independent interest and related to the
equivalence between tight 2-matroids and even delta-matroids observed by
Bouchet.Comment: 35 pages, 3 figure
Binary matroids and local complementation
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph
G. M(IAS(G)) classifies G up to local equivalence, and determines the
delta-matroid and isotropic system associated with G. Moreover, a parametrized
form of its Tutte polynomial yields the interlace polynomials of G.Comment: This article supersedes arXiv:1301.0293. v2: 26 pages, 2 figures. v3
- v5: 31 pages, 2 figures v6: Final prepublication versio
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