The Penultimate Rate of Growth for Graph Properties
Given a property sript P sign of graphs, write sript P signn for the set of graphs with vertex set [n] having property sript P sign. We call \sript P signn| the speed of sript P sign. Recent research has shown that the speed of a monotone or hereditary property sript P sign can be a constant, polynomial, or exponential function of n, and the structure of the graphs in sript P sign can then be well described. Similarly, \sript P signn\ can be of the form n(1-1/k+o(1))n or 2(1-1/k+o(1))n2/2 for some positive integer k > 1 and the properties can be described and have well-behaved speeds. In this paper, we discuss the behavior of properties with speeds between these latter bounds, i.e., between n(1+o(1))n and 2(1/2+o(1))n2/2. © 2001 Academic Press.
European Journal of Combinatorics
Balogh, J., Bollobás, B., & Weinreich, D. (2001). The Penultimate Rate of Growth for Graph Properties. European Journal of Combinatorics, 22 (3), 277-289. https://doi.org/10.1006/eujc.2000.0476