Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems

Counting in hypergraphs via regularity inheritance

We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma