Extremals of Functions on Graphs with Applications to Graphs and Hypergraphs

AbstractThe method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math. 17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turán's Theorem [Colloq. Math. 3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok 48 (1941), 436–452]

Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures

AbstractWe consider extremal problems ‘of Turán type’ for r-uniform ordered hypergraphs, where multiple oriented edges are permitted up to multiplicity q. With any such ‘(r, q)-graph’ Gn we associate an r-linear form whose maximum over the standard (n − 1)-simplex in Rn is called the (graph-) density g(Gn) of Gn. If ex(n, L) is the maximum number of oriented hyperedges in an n-vertex (r, q)-graph not containing a member of L, limn→∞ ex(n, L)/nr is called the extremal density of L. Motivated, in part, from results for ordinary graphs, digraphs, and multigraphs, we establish relations between these two notions

Endre Szemerédi, Premi Abel 2012

Aquest article presenta una breu descripció de les contribucions matemàtiques més destacades d'Endre Szemerédi, Premi Abel 2012.This article presents a short description of the main mathematical contributions of Endre Szemerédi, Abel Prize 2012

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio