Locally bounded coverings and factorial properties of graphs

For a graph property XX, let XnXn be the number of graphs with vertex set {1,…,n}{1,…,n} having property XX, also known as the speed of XX. A property XX is called factorial if XX is hereditary (i.e. closed under taking induced subgraphs) and nc1n≤Xn≤nc2nnc1n≤Xn≤nc2n for some constants c1c1 and c2c2. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. Only the properties with speeds up to the Bell number are well studied and well behaved. To better understand the behavior of factorial properties with faster speeds we introduce a structural tool called locally bounded coverings and show that a variety of graph properties can be described by means of this tool.