For a fixed integer \(k\geqslant 2\) , let \(G\in \mathcal {G}(n,p)\) be a simple connected graph on \(n\rightarrow \infty \) vertices with \(p= \frac{c}{n}\) for a large enough constant c. Any collection of edges \(M_k\) in G with the additional constraint that no two edges are within distance k, is called a distance k-matching. The k-matching number, denoted by \(um_k(G)\) , is the size of the largest distance k-matching \(M_k\) in G. Kang and Manggala showed that \(um_k(G)\leqslant (1+o(1))\frac{k n\log c}{2c^{k-1}}\) when \(k\geqslant 2\) and speculated that the upper bound is close to the correct value of \(um_k(G)\) . Cooley et al. confirmed this conclusion when \(k=2\) . Unfortunately, the approach does not work when \(k\geqslant 3\) . In this paper, we show that the size of any maximal distance k-matching in G asymptotically lies between \( \frac{(k-1)n\log c}{4c^{k-1}}\) and \( \frac{k n \log c}{2c^{k-1}}\) , and we also design a randomized greedy algorithm to generate one large distance k-matching in G with asymptotical size \( \frac{kn\log c}{4c^{k-1}}\) when \(k\geqslant 3\) . These results mean that \(um_k(G)\geqslant (1+o(1))\frac{kn\log c}{4c^{k-1}}\) when \(k\geqslant 3\) .
