By Alspach B., Xu M.Y.
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Extra resources for 1/2-Transitive Graphs of Order 3p
On the other hand, what is surprising is that one can get much global information about G(n, c/n) from the branching process. The reason why this is surprising is that the connection between the exploration process and the branching process must break down: eventually, in the graph we run out of vertices, while the branching process (for c > 1) may well continue forever. Moreover, for c > 1, the graph contains many cycles: in the branching process we never see these. Depending on how close a connection we want, the local approximation may well break down quite soon; certainly we cannot push the coupling argument as far as k = Θ(n) without losing 44 B.
Thus our results are consistent with, and extend, Theorem 4. On the other hand, in Theorem 5, whenever ε log n is unbounded, the diﬀerence is important, and to obtain a correct result one must replace 2ε−2 by δ −1 . ) As we shall see, δ plays a natural role in the branching process: it is the decay constant in the exponential tail probability that the process has large (but ﬁnite) total size. In the next two subsections we turn to our new proofs of Theorems 6 and 7, based on branching processes.
Proof of Theorem 7. Let (31) A1 = log L A0 = log L − (5/2) log log L − K, and where K = K(n) → ∞ but K ≤ (1/3) log log L, say, so A0 → ∞. Note that (32) −5/2 −A0 LA0 e ∼ eK → ∞, while (33) −5/2 −A1 LA1 e = (log L)−5/2 → 0. For i = 0, 1, let ki = Ai /δ, where δ = δ(1 − ε) is deﬁned by (21), and let S+ = kTk k0 ≤k≤k1 58 B. Bollob´ as and O. Riordan be the number of vertices of Gn = G(n, (1 − ε)/n) in tree components of order between k0 and k1 . The required lower bound follows if we can show that S+ ≥ 1 whp.