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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.