The property of having a k ‐regular subgraph has a sharp threshold

Abstract

We prove that the property of containing a k-regular subgraph in the random graph model G(n, p) has a sharp threshold for k ≥ 3. We also show how to use similar methods to obtain an easy prove for the (known fact of) sharpness of having a non empty k-core for k ≥ 3.

Threshold functions for small subgraphs: an analytic approach

“…Several papers have studied the relation between a k-regular subgraph and a k-core: Pra lat, Verstraëte and Wormald [39] prove that the threshold for the appearance of a k-regular subgraph is at most the threshold for the appearance of a non-empty (k + 2)-core, a result improved first by Chan and Molloy [10] to a (k + 1)-core, and further by Gao [20], who showed that the size of a k-regular subgraph is "close" to the size of the k-core. Concurrently, Letzter [31] has obtained the existence of a sharp threshold for the existence of a k-regular subgraph for k ≥ 3. Very recently, Mitsche, Molloy and Pra lat [36] proved that a random graph G(n, p = c/n) typically has a k-regular subgraph if c > e −Θ(k) , which is above the threshold for the appearance of a k-core.…”

The Firstk-Regular Subgraph is Large

“…, n}, where each edge occurs independently with probability p. This paper relates to the research on the existence of k-regular subgraphs of G n,cn/2 (correspondingly G(n, c/n)), when c is slightly greater than c k , the threshold of the emergence of the k-core. It was shown in [8] that the property of having a k-regular subgraph has a sharp threshold p * (n, k) in G(n, p) for all k ≥ 3, where p * (n, k) is of order Θ(1/n). Whether the limit of np * (n, k) exists is not known.…”

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

“…Set Z to be the number of other copies of u that are paired with vertex-copies in W 0 . Note that E(Z) is the LHS of (10).…”