On Induced Paths, Holes, and Trees in Random Graphs

The concentration of the sizes of largest induced paths and cycles (holes) is studied in Erdős–Rényi random graphs. A 2-point concentration is proved for the size of the largest induced path and cycle for all p=p(n) satisfying p≥n−1/2(lnn)2 and p≤1−ε , where ε>0 is any constant. No such tight concentration (within two consecutive values) was previously known for induced paths and cycles. As a corollary, a significant additive improvement is obtained over a 40-year-old result of Erdős and Palka [Discrete Math., 46 (1983), pp. 145–150] concerning the size of the largest induced tree in a dense random graph. Further, the induced path decomposition number and induced tree decomposition number, i.e., the smallest number of parts into which the vertex set of a graph can be partitioned such that every part induces a (i) path or (ii) tree, respectively, are studied for G(n,p) . The arguments involve the second moment method together with an adaptation of a martingale-based technique of Krivelevich et al. [Random Structures Algorithms, 22 (2003), pp. 1–14] for monotone high-degree polynomial random variables to the nonmonotone setting. A lower bound is proved showing the tightness of the application of the inequality up to logarithmic factors in the exponent. The modified inequality is then stated and proved in a general setting, which may be of independent interest.