Distance-2-Matchings of Random Graphs

Abstract

In this paper we study the maximal size of a distance-2-matching in a random graph G n;M , i.e., the probability space consisting of subgraphs of the complete graph over n vertices, K n , having exactly M edges and uniform probability measure. A distance-2-matching in a graph Y, M 2, is a set of Y-edges with the property that for any two elements every pair of their 4 incident vertices has Y-distance ≥ 2. Let M2(Y) be the maximal size of a distance-2-matching in Y. Our main results are the derivation of a lower bound for M2(Y) and a sharp concentration result for the random variable $$ \mathrm{M}_2 : G_{n,M} \rightarrow \mathbb{Z} \quad \mathrm{for} \quad M = c(n-1)/2 \quad \mathrm{with} \quad c > 0 $$