Abstract. We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an efficient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G −→ H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G −→ H.
This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány
Abstract. This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization. The survey contains several open problems.
We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures µ1, µ2, .. . , C k such that µi(Cj) = 1 for all i, j. If k = 2 this gives the ham sandwich Theorem. This result was independently proved by R.N. Karasev.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.