15 research outputs found
Seymour's second neighborhood conjecture for tournaments missing a generalized star
Seymour's Second Neighborhood Conjecture asserts that every digraph (without
digons) has a vertex whose first out-neighborhood is at most as large as its
second out-neighborhood. We prove its weighted version for tournaments missing
a generalized star. As a consequence the weighted version holds for tournaments
missing a sun, star, or a complete graph.Comment: Accepted for publication in Journal of Graph Theory in 24 June 201
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple dicycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but have a vertex
of minimum outdegree that has the Second Neighborhood Property.Comment: fixed an error in an earlier version and made structural change
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple cycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but that local
simplicity can, in some cases, also imply the existence of a Seymour vertex in
the original digraph.Comment: This is the version accepted for publication in Opuscula Mathematic
A Remark on the Second Neighborhood Problem
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.Comment: arXiv admin note: substantial text overlap with arXiv:1106.546
A contribution to the second neighborhood problem
Seymour's Second Neighborhood Conjecture asserts that every digraph (without
digons) has a vertex whose first out-neighborhood is at most as large as its
second out-neighborhood. It is proved for tournaments, tournaments missing a
matching and tournaments missing a generalized star. We prove this conjecture
for classes of digraphs whose missing graph is a comb, a complete graph minus 2
independent edges, or a complete graph minus the edges of a cycle of length 5