On the probability of connectedness
نویسنده
چکیده
Given a class of structures with a notion of connectedness (satisfying some reasonable assumptions), we consider the limit (as n -+ oc) of the probability that a random (labelled or unlabelled) n-element structure in the class is connected. The paper consists of three parts: two specific examples, N-free graphs and posets, where the limiting probability of connectedness is one-half and the golden ratio respectively; an investigation of the relation between this question and the growth rate of the number of structures in the class; and a generalisation of the problem to other combinatorial constructions motivated in part by the group-theoretic constructions of direct and wreath product. 1. An example: N-free posets In this section, certain graphs and posets will be considered. The notion of connectedness of a graph is clear, while a poset is connected if its comparability graph is connected. Given a class ~ of graphs or posets, we select at random an isomorphism class of n-element structures in g (the unlabelled case) or a structure on {1,2 . . . . . n} isomorphic to a structure in cg (the labelled case); let p,, be the probability that the chosen structure is connected. Our general question is: Does lim . . . . (Pn) exist, and i f so what is" it? The symbol N will denote the graph or the poset which is shown in Fig. 1. A graph or poser is called N-free if it does not contain N as an induced substructure. The class of N-free graphs has been studied in many contexts, under many different names. I summarise the main facts. The complement of an N-free graph is N-free. An N-free graph with more than one vertex is connected if and only if its complement is disconnected. * E-mail: [email protected]. 0012-365X/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH SOOI2-365X(96)O0226-9 176 P.J. Cameron / Discrete Mathematics 167/168 (1997) 175 187
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ورودعنوان ژورنال:
- Discrete Mathematics
دوره 167-168 شماره
صفحات -
تاریخ انتشار 1997