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The Proportion of Unlabelled Graphs which are Hamiltonian

The Proportion of Unlabelled Graphs which are Hamiltonian THE PROPORTION OF UNLABELLED GRAPHS WHICH ARE HAMILTONIAN E. M. WRIGHTj An («, q) graph has n nodes and q edges, but no slings or multiple edges. We write N = \n{n-1) (so that 0 < q ^ N), B(h, k) = h\/{k\(h-k)\}, the usual binomial coefficient, F = F(n,q) for the number of labelled (n,q) graphs and T = T(n,q) for the number of unlabelled (n, q) graphs. Clearly F = B(N, q). We write also p = p(n,q) for the proportion of labelled (n,q) graphs which are Hamiltonian and a = a(n, q) for the corresponding proportion of unlabelled (w, q) graphs. Any statement involving C is true for some fixed positive number C (not always the same) independent of n and 4. We shall prove the following result. THEOREM 1. As n -*• 00, we have sup|p — a\ -*0. (1) It is reasonably obvious that, for fixed n, the number p(n,q) increases with q, at least in the non-strict sense. (The proof is similar to that in [11] for the corresponding result for connectedness). One might expect the same to be true for a, but it is not so. THEOREM 2. For sufficiently large n, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Proportion of Unlabelled Graphs which are Hamiltonian

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/8.3.241
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Abstract

THE PROPORTION OF UNLABELLED GRAPHS WHICH ARE HAMILTONIAN E. M. WRIGHTj An («, q) graph has n nodes and q edges, but no slings or multiple edges. We write N = \n{n-1) (so that 0 < q ^ N), B(h, k) = h\/{k\(h-k)\}, the usual binomial coefficient, F = F(n,q) for the number of labelled (n,q) graphs and T = T(n,q) for the number of unlabelled (n, q) graphs. Clearly F = B(N, q). We write also p = p(n,q) for the proportion of labelled (n,q) graphs which are Hamiltonian and a = a(n, q) for the corresponding proportion of unlabelled (w, q) graphs. Any statement involving C is true for some fixed positive number C (not always the same) independent of n and 4. We shall prove the following result. THEOREM 1. As n -*• 00, we have sup|p — a\ -*0. (1) It is reasonably obvious that, for fixed n, the number p(n,q) increases with q, at least in the non-strict sense. (The proof is similar to that in [11] for the corresponding result for connectedness). One might expect the same to be true for a, but it is not so. THEOREM 2. For sufficiently large n,

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1976

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