By Fiorini, Stanley; Wilson, Robin J
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Additional resources for Edge colourings of graphs
In fact, the motivation for considering kernels (graphons) in the dense case was not so much to deﬁne a new random graph model, as to understand limits of sequences of (deterministic) graphs; this program was carried out by Borgs, Chayes, Lov´asz, S´ os, Szegedy and Vesztergombi in a series of papers [56, 55, 141, 142, 57, 58]; we return to this in Section 6. Of course, one can pass from the dense graph G1 (n, κ) to (a special case of) the sparse graph G1/n (n, κ) by deleting edges, keeping each with probability c/n.
If p = c/n with c > 0 constant and k is ﬁxed, this probability is o(1), and it follows from Lemma 1 that (4) 1 E Nk0 (Gn ) → P ( X(c) = k ). n Although this distinction is not always made, there is in principle a big diﬀerence between E Nk (Gn ) /n and what we would really like to study: the fraction of vertices in components of order k. This fraction is a random quantity, and it could a priori vary a lot, and so need not be close to its expectation. However, there is a simple trick to show that this is not the case here, requiring essentially no further work.
53 Random Graphs and Branching Processes When √ k → ∞, then using Stirling’s formula in the (rather crude) form k! ∼ 2πk k+1/2 e−k , we may rewrite (19) as (20) ρk (λ) ∼ (2π)−1/2 k−3/2 λ−1 λe1−λ k . The quantity λe1−λ turns out to play a fundamental role in the analysis of X(λ) or of G(n, λ/n). It is easily seen that λe1−λ is at most 1, so it is convenient to consider the negative of its logarithm. Thus we set (21) δ = δ(λ) = − log λe1−λ = λ − 1 − log λ. In this notation, recalling that the approximation in Stirling’s formula is correct within 10% for all k ≥ 1, we have (22) ρk (λ) ≤ k−3/2 λ−1 e−δk for all k and λ.
Edge colourings of graphs by Fiorini, Stanley; Wilson, Robin J