215 research outputs found
Note on Disjoint Cycles in Multipartite Tournaments
In 1981, Bermond and Thomassen conjectured that for any positive integer ,
every digraph with minimum out-degree at least admits
vertex-disjoint directed cycles. In this short paper, we verify the
Bermond-Thomassen conjecture for triangle-free multipartite tournaments and
3-partite tournaments. Furthermore, we characterize 3-partite tournaments with
minimum out-degree at least () such that in each set of
vertex-disjoint directed cycles, every cycle has the same length.Comment: 9 pages, 0 figur
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
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