The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and colored labeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity 0 was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colored labeled rooted trees of activity 0 .

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z^d, the lists are order ideals in the integer lattice Z^d, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n x d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.

We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.

We study the set of NBC sets (no broken circuit sets) of the Linial arrangement and deduce a constructive bijection to the set of local binary search trees. We then generalize this construction to two families of Linial type arrangements for which the bijections are with some $k$-ary labelled trees that we introduce for this purpose.

In this paper we discuss Alpha Galois lattices (Alpha lattices for short) and the corresponding association rules. An alpha lattice is coarser than the related concept lattice and so contains fewer nodes, so fewer closed patterns, and a smaller basis of association rules. Coarseness depends on a a priori classification, i.e. a cover \({\mathcal C}\) of the powerset of the instance set I, and on a granularity parameter α. In this paper, we define and experiment a Merge operator that when applied to two Alpha lattices \(G({\mathcal C}_1,\alpha) \) and \(G({\mathcal C}_2,\alpha)\) generates the Alpha lattice \(G({\mathcal C}_1 \cup {\mathcal C}_2,\alpha)\), so leading to a class-incremental construction of Alpha lattices. We then briefly discuss the implementation of the incremental process and describe the min-max bases of association rules extracted from Alpha lattices.

Let ck(G) be the minimum number of elementary cycles of length at most k necessary to cover the vertices of a given graph G. In this work, we bound ck(G) by a function of the order of G and its independence number.

The main content of the note is a proof of the conjecture of Hamidoune and Las Vergnas on the directed switching game in the case of Lawrence oriented matroids.

A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are analogous to those of the chromatic polynomial of an ordinary graph, though they are different from those usually assumed of gain graphs or matroids. The three laws lead to the weak chromatic group of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for all switching-invariant functions of gain graphs, such as chromatic polynomials, that satisfy the deletion-contraction identity for neutral links and are zero on graphs with neutral loops. Examples are the total chromatic polynomial of any gain graph, including its specialization the zero-free chromatic polynomial, and the integral and modular chromatic functions of an integral gain graph. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. The proof involves gain graphs between the Catalan and Shi graphs whose polynomials are expressed in terms of descending-path vertex partitions of the graph of $(-1)$-gain edges. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.

A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological hyperplanes in H, if their intersection is nonempty, meet in a subspace that is a topological hyperplane in the intersection of any k-1 of them; but two topological hyperplanes that do intersect need not cross each other. If every intersecting pair does cross, the arrangement is affine. The number of regions formed by an arrangement of topological hyperplanes has the same formula as for arrangements of affine hyperplanes. Hoping to explain this geometrically, we ask whether parts of the topological hyperplanes in any arrangement can be reassembled into an arrangement of affine topological hyperplanes with the same regions. That is always possible if the dimension is two but not in higher dimensions. We also ask whether all affine topological hyperplane arrangements correspond to oriented matroids; they need not, but we can characterize those that do if the dimension is two. In higher dimensions this problem is open. Another open question is to characterize the intersection semilattices of topological hyperplane arrangements; a third is to prove that the regions of an arrangement of topological hyperplanes are necessarily cells.

Jürgen Bokowski and Tomaž Pisanski Oriented matroids and complete-graph embeddings on surfaces … Balázs Szegedy Coverings of Abelian groups and vector spaces … Cai Heng Li and Ákos Seress Symmetrical path-cycle covers of a graph and polygonal graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  … Binod Kumar Sahoo and NS Narasimha Sastry … Winfried Bruns and Tim Römer h-Vectors of Gorenstein polytopes … Guangguo Han and Huiling Li Unsolvable block transitive automorphism groups of 2-(v,5,1) designs . . . . . . . . . . . . . . . . . . . .  … David Forge and Thomas Zaslavsky Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements . . . . . . . . . . . . . . . .  … Michael Giudici, Cai Heng Li, Cheryl E. Praeger, Ákos Seress, and …

Given a graph G with n vertices, we call ck(G) the minimum number of elementary cycles of length at most k necessary to cover the vertices of G. We bound ck(G) from the minimum degree and the order of the graph.

We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field.

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.

We present dichromatic and tree-expansion polynomials of integral gain graphs that underlie the problem of counting lattice points in the complement of an integral anographic hyperplane arrangement. This is a step towards finding the universal Tutte invariant of rooted integral gain graphs. Mathematics Subject Classifications (2000): Primary 05C22; Secondary 05C15.

A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.

To encode an important property of the “no broken circuit bases” of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called ”diagonal basis.” We prove that this definition extends naturally to a large class of algebras, the so called χ- algebras. Our definitions make also use of an “iterative residue formula” based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.

In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the no broken circuit of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called bases. This notion extends naturally to X-algebras. We give a survey of the results obtained by the authors concerning the construction of Groebner bases of I(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an ``iterative residue formula'' introduced by Szenes.