2 József Balogh , Béla Bollobás , And
نویسنده
چکیده
A collection of unlabelled tournaments P is called a hereditary property if it is closed under isomorphism and under taking induced sub-tournaments. The speed of P is the function n 7→ |Pn|, where Pn = {T ∈ P : |V (T )| = n}. In this paper, we prove that there is a jump in the possible speeds of a hereditary property of tournaments, from polynomial to exponential speed. Moreover, we determine the minimal exponential speed, |Pn| = c , where c ≃ 1.47 is the largest real root of the polynomial x 3 = x + 1, and the unique hereditary property with this speed.
منابع مشابه
The structure of almost all graphs in a hereditary property
A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n 7→ |Pn|, where Pn denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then |Pn| = 2 2/2, where r = r(P) ∈ N is the so-called ‘colouring number’ of P . However, their result...
متن کاملProbability HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbour model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.
متن کاملHamilton Cycles in Random Geometric Graphs
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This proves a conjecture of Penrose. We also show that in the k-nearest neighbour model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.
متن کاملMeasures on monotone properties of graphs
Given a monotone property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. Building on recent results in the enumeration of graphical properties, we prove numerous results about the structure of graphs in Pn and the functions |Pn|. We also examine the measure eP (n), the maximum number of edges in a graph of Pn.
متن کامل