Triangle resilience of the square of a Hamilton cycle in random graphs

Since first introduced by Sudakov and Vu in 2008, the study of resilience problems random graphs received a lot attention probabilistic combinatorics. Of particular interest are spanning structures. It is known that for structures which contain many triangles, local cannot prevent an adversary from destroying all copies structure removing negligible amount edges incident to every vertex. In this paper we generalise notion H -resilience demonstrate its usefulness on containment problem square Hamilton cycle. particular, show there exists constant C > 0 such if p ? log 3 ? n / then w.h.p. subgraph G graph , cycle, provided vertex remains at least ( 4 9 + o 1 ) -fraction triangles . The 4/9 optimal value slightly improves best-known appearance threshold up logarithmic factor.