Large Induced Matchings in Random Graphs

Strong and weighted matchings in inhomogenous random graphs

“…with high probability, where we recall that k 1 is roughly equal to k 2 . Extrapolating the results of [6] for induced matchings where k = 1, we conjecture that ν k (G) is in fact of the order of n(log n) a n k(1−β+δ) with high probability, for some constant a > 0.…”

“…Induced matchings have been studied in [5] (see also references therein) under the name of strong matchings where estimates for the expected size of a maximum induced matching in homogenous graphs with constant edge probability is obtained. Recently [6] used a combination of second moment method along with concentration inequalities to estimate the largest possible size of induced matchings in homogenous graphs and obtained deviation bounds for a wide range of edge probabilities. A main bottleneck in directly extending the above results to inhomogenous graphs is that induced matchings do not satisfy the monotonicity property enjoyed by the ordinary matchings described in the previous paragraph.…”

Large induced distance matchings in certain sparse random graphs

“…Recently, Cooley et al in [2] confirmed this conclusion when k = 2, d c and d = o(n) for some large enough constant c. They showed that um 2 (G) = (1 + o(1)) n log d d relying on two main ingredients: the second moment method and Talagrand's inequality. Talagrand's inequality is a useful tool to show that a random variable is tightly concentrated under certain conditions.…”

“…We often refer to a path by the natural sequence of its vertices, writing P = x 0 x 1 • • • x k with distinct vertices x i for 0 i k and calling P a path between x 0 and x k of length k. The qualifier "distance" is normally omitted in the remainder of this paper. A 1-matching is a matching and a 2-matching is also known as an induced matching or a strong matching [2,4,6]. A k-matching M k is said to be maximal if no other edges can be added to M k while keeping the property of being a distance k away.…”

“…In this paper, for any fixed integer k 2 and G ∈ G(n, p), as a further step of [2,3,5,7], we consider the bounds of um k (G). In fact, we mainly investigate e(M k ) of any maximal k-matching M k in G ∈ G(n, p).…”