Seymour's Second Neighborhood Conjecture (SSNC) asserts that every oriented finite simple graph (without digons) has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). In this paper, we prove SSNC for tournaments missing two stars. We also study SSNC for tournaments missing disjoint paths and, particularly, in the case of missing paths of length 2. In some cases, we exhibit at least two vertices with the SNP.

Let C be an oriented cycle with two blocks and let n denote the number of its vertices. We show that every $(n+2)$-chromatic digraph spanned by a Hamiltonian directed path contains a subdivision of C. This bound is tight.

Let C be any oriented cycle on n vertices. We show that every $(10n-4)$-chromatic digraph spanned by a Hamiltonian directed path contains a subdivision of C.