On the number of vertexdisjoint cycles in digraphs
Abstract
Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k1$ contains $k$ vertexdisjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, SpringerVerlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. BangJensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284302] that every digraph with girth $g$ and minimum outdegree at least $\frac{g}{g1}k$ contains $k$ vertexdisjoint cycles. Thomassé conjectured around 2005 that every oriented graph (a digraph without 2cycles) with girth $g$ and minimum outdegree at least $h$ contains a path of length $h(g1)$, where $h$ is a positive integer. In this note, we first present a new shorter proof of the BermondThomassen conjecture for the case of $k=3$, and then we disprove the conjecture proposed by BangJensen, Bessy and Thomassé. Finally, we disprove the even girth case of the conjecture proposed by Thomassé.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.02999
 Bibcode:
 2018arXiv180502999B
 Keywords:

 Mathematics  Combinatorics