1,230 research outputs found
Hopf algebraic structures on mixed graphs
We introduce two coproducts on mixed graphs (that is to say graphs with both
edges and arcs), the first one by separation of the vertices into two parts,
and the second one given by contraction and extractions of subgraphs. We show
that, with the disjoint union product, this gives a double bialgebra, that is
to say that the first coproduct makes it a Hopf algebra in the category of righ
comodules over the second coproduct. This structures implies the existence of a
unique polynomial invariants on mixed graphs compatible with the product and
both coproducts: we prove that it is the (strong) chromatic polynomial of Beck,
Bogart and Pham. Using the action of the monoid of characters, we relate it to
the weak chromatic polynomial, as well to Ehrhart polynomials and to a
polynomial invariants related to linear extensions. As applications, we give an
algebraic proof of the link between the values of the strong chromatic
polynomial at negative values and acyclic orientations (a result due to Beck,
Blado, Crawford, Jean-Louis and Young) and obtain a combinatorial description
of the antipode of the Hopf algebra 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
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 -vector.
We apply our results to the weak and strong chromatic polynomials of acyclic
mixed graphs, and the order polynomial of a double poset.Comment: 13 pages, extended abstract for FPSAC 202
Chromatic polynomials of some sunflower mixed hypergraphs
The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D-, C-, or B-hypergraph respectively, where B = C = D. D-hypergraph colorings are the classic hypergraph colorings which have been widely studied. The chromatic polynomial P(H;λ) of a mixed hypergraph H is the function that counts the number of proper λ-colorings, which are mappings. Recently, Walter published [15] some results concerning the chromatic polynomial of some non-uniform D-sunflower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunflowers and B-sunflowers. Some results of a new but related member of sunflowers are also presented
Bivariate Chromatic Polynomials of Mixed Graphs
The bivariate chromatic polynomial of a graph ,
introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all -colorings of
such that adjacent vertices get different colors if they are . We
extend this notion to mixed graphs, which have both directed and undirected
edges. Our main result is a decomposition formula which expresses
as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz
2020), and a combinatorial reciprocity theorem for .Comment: 10 pages, 3 figures, Revised according to referee comment
Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings
For a poset P, a subposet A, and an order preserving map F from A into the
real numbers, the marked order polytope parametrizes the order preserving
extensions of F to P. We show that the function counting integral-valued
extensions is a piecewise polynomial in F and we prove a reciprocity statement
in terms of order-reversing maps. We apply our results to give a geometric
proof of a combinatorial reciprocity for monotone triangles due to Fischer and
Riegler (2011) and we consider the enumerative problem of counting extensions
of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3:
examples included (suggested by referee), to appear in "SIAM Journal on
Discrete Mathematics
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