Decision trees, monotone functions, and semimatroids

We define decision trees for monotone functions on a simplicial complex. We define homology decidability of monotone functions, and show that various monotone functions related to semimatroids are homology decidable. Homology decidability is a generalization of semi-nonevasiveness, a notion due to Jonsson. The motivating example is the complex of bipartite graphs, whose Betti numbers are unknown in general. We show that these monotone functions have optimum decision trees, from which we can compute relative Betti numbers of related pairs of simplicial complexes. Moreover, these relative Betti numbers are coefficients of evaluations of the Tutte polynomial, and every semimatroid collapses onto its broken circuit complex.Comment: 16 page

The Chromatic Quasisymmetric Class Function of a Digraph

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the former is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including F-positivity, and combinatorial reciprocity theorems. We deduce results for orbital quasisymmetric functions, and study a generalization of the notion of strongly flawless sequences

Coloring complexes and combinatorial Hopf monoids

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids

On Equivariant flag f-vectors for balanced relative simplicial complexes

We study the equivariant flag f-vector and equivariant flag h-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre\u27s condition (Sℓ), we show that the equivariant flag h-vector, the equivariant h-vector, and the equivariant f-vector satisfy several inequalities.We apply these results to the study of P-partitions of double posets, and weak colorings of mixed graphs

On Cohen-Macaulay Hopf monoids in species

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen- Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative h-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset

Brooks\u27 theorem for 2-fold coloring

The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph. We prove that, for 2-connected graphs, the two-fold chromatic number is at most the Ore degree, unless G is a complete graph or an odd cycle