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Article

Locally bounded coverings and factorial properties of graphs

European Journal of Combinatorics. 2012. Vol. 33. No. 4. P. 534-543.
Lozin V. V., Mayhil C., Zamaraev V. A.

For a graph property X, let Xn be the number of graphs with vertex set {1, . . . , n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e. closed under taking induced subgraphs) and nc1n ≤ Xn ≤ nc2n for some constants c1 and c2. 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.