44 research outputs found
On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid
The first author introduced the circuit-cocircuit reversal system of an
oriented matroid, and showed that when the underlying matroid is regular, the
cardinalities of such system and its variations are equal to special
evaluations of the Tutte polynomial (e.g., the total number of
circuit-cocircuit reversal classes equals , the number of bases of
the matroid). By relating these classes to activity classes studied by the
first author and Las Vergnas, we give an alternative proof of the above results
and a proof of the converse statements that these equalities fail whenever the
underlying matroid is not regular. Hence we extend the above results to an
equivalence of matroidal properties, thereby giving a new characterization of
regular matroids.Comment: 7 pages. v2: simplified proof, with new statements concerning other
special evaluations of the Tutte polynomia
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
On the chromatic number of cube-like graphs
AbstractA cube-like graph is a graph whose vertices are all 2n subsets of a set E of cardinality n, in which two vertices are adjacent if their symmetric difference is a member of a given specified collection of subsets of E. Many authors were interested in the chromatic number of such graphs and thought it was always a power of 2. Although this conjecture is false (we show a cube-like graph of chromatic number 7), we prove that there is no cube-like graph with chromatic number 3