A remark on the second neighborhood problem

Salman Ghazal (Department of Mathematics Faculty of Sciences I, Lebanese University Beirut - Lebanon)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 3, No 2 (2015): Electronic Journal of Graph Theory and Applications

Seymour's second neighborhood conjecture states that every simple digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). We define "good" digraphs and prove a statement that implies that every feed vertex of a tournament has the SNP. In the case of digraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due to Fidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices with SNP