The Firstk-Regular Subgraph is Large

Abstract

Let (G m ) 0 m ( n 2 ) be the random graph process starting from the empty graph on vertex set [n] and with a random edge added in each step. Let m k denote the minimum integer such that G m k contains a k-regular subgraph. We prove that for all sufficiently large k, there exist two constants k σ k > 0, with k → 0 as k → ∞, such that asymptotically almost surely any k-regular subgraph of G m k has size between (1 − k )|C k | and (1 − σ k )|C k |, where C k denotes the k-core of G m k .

k-regular subgraphs near the k-core threshold of a random graph

“…In a nutshell, the observation follows by noting that, since STRIP2 carries out a very small number of iterations and hence deletes a very small number of vertices, these parameters will not change much from their initial values as provided in Lemma 12. In more detail, STRIP2 runs for at most βn iterations, where β = e −k/200 from (8), and in each iteration we delete one vertex.…”

“…Part (a) is well-known; the size of the k-core approaches n as k grows (see, for example, the results of Molloy [12] or Gao [8]). Indeed, in [12] it is proved that w.h.p.…”

“…Our final piece is the deferred proof regarding properties of the subgraph K obtained by our stripping procedure. Recall our setting: we begin with the random graph G = G n,p=c/n where c k + k 10 β ≤ c ≤ c k + k −1/2 (see (8), (9)). K is the subgraph of the k-core of G obtained after applying STRIP, and possibly deleting one more vertex.…”

Threshold functions for small subgraphs: an analytic approach

“…Since L n (z, x, t) is uniformly bounded, by Condition (C3) truncating the circle |x| = 1 to |x| = 1, arg(x) ≤ ε in (20) causes an asymptotically negligible error. Next, we start with the evaluation of the inner integral.…”

On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time

“…Mitsche, Molloy and Pralat [14] proved that for sufficiently large k, ψ k − c k ≤ e −k/300 while Gao [9] proved that if |p − ψ k /n| is sufficiently small then any k-regular subgraph of G n,p must contain all but at most ǫ k n vertices of the k-core of G n,p for some ǫ k that tends to 0 as k → ∞.…”