An elementary chromatic reduction for gain graphs and special hyperplane arrangements

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 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 chromatic functions of gain graphs. 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. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page

A combinatorial perspective on the non-Radon partitions

AbstractLet E be a finite set of points in Rd. Then {A, E − A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E−A. Or equivalently iff AO is an acyclic reorientation of (MAff(E), O), the oriented matroid canonically determined by E. If (M(E), O) is an oriented matroid without loops then the set NR(E, O) = {(A, E − A): AO is acyclic} determines (M(E), O). In particular the matroidal properties of a finite set of points in Rd are precisely the properties which can be formulated in non-Radon partitions terms. The Möbius function of the poset A = {A: A ⊆ E, AO is acyclic} and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of Rn dissected by hyperplane

Polarity and point extensions in oriented matroids

AbstractA. Bachem and W. Kern have recently extended the notion of polarity (relatively to an R-bilinear form) to oriented matroids [1]. We prove that the usual polarity properties of the face lattices of convex polytopes can be extended to the class of oriented matroids admitting an (oriented) polar. We give also a short proof of the principal result of [1] showing that there is a natural embedding of the poset of signed span of the cocircuits of a polar of an oriented matroid into the extension poset of this matroid. We remark that if M is a matroid admitting a polar, then every hyperplane can be intersected by every line. Oriented matroids satisfying this condition have an important role in oriented-matroid programming

The Fundamental Group of the Complement of the Complexification of a Real Arrangement of Hyperplanes

AbstractLet A be an arrangement of hyperplanes (i.e., a finite set of 1-codimension vector subspaces) in Rd. We say that the linear ordering of the hyperplanes A,H1≺H2≺···≺Hn, is ashellabilityorder of A, if there is an oriented affine linelcrossing the hyperplanes of A on the given linear order. The intersection latticeL(A) is the set of all intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(Ac)≔Cd\⋃{H⊗C:H∈A}. We will prove:Suppose that there are shellability orders H1≺H2≺···≺Hnand H′1≺′H′2≺′···≺′H′n, respectively, ofAandA′,such that the bijective map Hi→H′i, i∈[n]determines an isomorphism of the intersection lattices L(A)and L(A′).Then the fundamental groupsπ1(M(Ac))andπ1(M(A′c)) are isomorphic

How is a Chordal Graph like a Supersolvable Binary Matroid?

Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.Comment: 10 pages, 3 figures, to appear in Discrete Mathematic