An Ehrhart theoretic approach to generalized Golomb rulers

A Golomb ruler is a sequence of integers whose pairwise differences, or equivalently pairwise sums, are all distinct. This definition has been generalized in various ways to allow for sums of h integers, or to allow up to g repetitions of a given sum or difference. Beck, Bogart, and Pham applied the theory of inside-out polytopes of Beck and Zaslavsky to prove structural results about the counting functions of Golomb rulers. We extend their approach to the various types of generalized Golomb rulers.Comment: 15 pages, 2 figure

Bivariate Chromatic Polynomials of Mixed Graphs

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.Comment: 10 pages, 3 figures, Revised according to referee comment

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

Global Constraint Catalog, 2nd Edition

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

Global Constraint Catalog, 2nd Edition (revision a)

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms