Local resilience of an almost spanning $k$-cycle in random graphs

The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any $k \geq 2$, every graph on $n$ vertices with minimum degree $kn/(k + 1)$ contains the $k$-th power of a Hamilton cycle. We extend this result to a sparse random setting. We show that for every $k \geq 2$ there exists $C > 0$ such that if $p \geq C(\log n/n)^{1/k}$ then w.h.p. every subgraph of a random graph $G_{n, p}$ with minimum degree at least $(k/(k + 1) + o(1))np$, contains the $k$-th power of a cycle on at least $(1 - o(1))n$ vertices, improving upon the recent results of Noever and Steger for $k = 2$, as well as Allen et al. for $k \geq 3$. Our result is almost best possible in three ways: for $p \ll n^{-1/k}$ the random graph $G_{n, p}$ w.h.p. does not contain the $k$-th power of any long cycle; there exist subgraphs of $G_{n, p}$ with minimum degree $(k/(k + 1) + o(1))np$ and $\Omega(p^{-2})$ vertices not belonging to triangles; there exist subgraphs of $G_{n, p}$ with minimum degree $(k/(k + 1) - o(1))np$ which do not contain the $k$-th power of a cycle on $(1 - o(1))n$ vertices.