Large induced distance matchings in certain sparse random graphs

Abstract

For a fixed integer k 2, let G ∈ G(n, p) be a simple connected graph on n → ∞ vertices with the expected degree d = np satisfying d c and d k−1 = o(n) for some large enough constant c. We show that the asymptotical size of any maximal collection of edges M in G such that no two edges in M are within distance k, which is called a distance k-matching, is between (k−1)n log d 4d k−1 and kn log d 2d k−1 . We also design a randomized greedy algorithm to generate one large distance k-matching in G with asymptotical size kn log d 4d k−1 . Our results partially generalize the results on the size of the largest distance k-matchings from the case k = 2 or d = c for some large constant c.