The largest hole in sparse random graphs

Abstract

We show that for any d=d(n) with d0(ϵ)≤d=o(n), with high probability, the size of a largest induced cycle in the random graph G(n,d/n) is (2±ϵ)ndlogd. This settles a long‐standing open problem in random graph theory.

Short proofs for long induced paths

“…The classical literature on Random Graphs considers many problems on finding induced subgraphs, starting from several independent papers [4,21,28] in the 1970's calculating the asymptotic independence number of G(n, p) for fixed p and large n. This was later extended by Frieze [17] to p = c/n for large constant c (as noted in [8], this extends to all p ≥ c/n). Erdős and Palka [16] initiated the study of induced trees in G(n, p), which developed a large literature [9,18,25,27] before its final resolution by de la Vega [10], showing that the size of the largest induced tree matches the asymptotics found earlier for the independence number: it is ∼ 2 log q (np) for all p ≥ c/n for large constant c. The Longest Induced Path problem in G(n, p) is also classical [26,31], and was recently resolved asymptotically by Draganić, Glock and Krivelevich [12].…”