Structural Properties of Sparse Graphs

Dense graphs have been extensively studied in the context of Extremal Graph Theory. The outstanding Szemerédi Regularity Lemma [111] states that any dense network has properties which are close to the ones of a random graph. In particular, a large dense network cannot be too irregular. This structural result is one of the cornerstones of contemporary combinatorics (and one would like to say mathematics in general). It also let to manyfold applications and generalizations, see e.g. [66, 65, 72, 113, 41]. The closest to our topic covered in this paper is the recent development which is based on the study of homomorphisms of graphs (and structures). (It is perhaps of interest note in how many different areas and a variety contexts the notion of a homomorphism recently appeared, see [60]). Regularity is viewed here as a sructural approximation in a proper metrics and also as a convergence. For a survey of this development see [15]. The main idea here is to study the local structure of a large graph G by counting the homomorphisms from various small graphs F into G (this relates to the area called property testing), and to study the global structure of G by counting its homomorphisms into various small graphs H (sometimes interpreted as templates). Very schematically this may be outlined by the schema: