On Hamiltonian Bypasses in one Class of Hamiltonian Digraphs
Abstract
Let $D$ be a strongly connected directed graph of order $n\geq 4$ which satisfies the following condition (*): for every pair of nonadjacent vertices $x, y$ with a common inneighbour $d(x)+d(y)\geq 2n1$ and $min \{ d(x), d(y)\}\geq n1$. In \cite{[2]} (J. of Graph Theory 22 (2) (1996) 181187)) J. BangJensen, G. Gutin and H. Li proved that $D$ is Hamiltonian. In [9] it was shown that if $D$ satisfies the condition (*) and the minimum semidegree of $D$ at least two, then either $D$ contains a preHamiltonian cycle (i.e., a cycle of length $n1$) or $n$ is even and $D$ is isomorphic to the complete bipartite digraph (or to the complete bipartite digraph minus one arc) with partite sets of cardinalities of $n/2$ and $n/2$. In this paper we show that if the minimum outdegree of $D$ at least two and the minimum indegree of $D$ at least three, then $D$ contains also a Hamiltonian bypass, (i.e., a subdigraph is obtained from a Hamiltonian cycle by reversing exactly one arc).
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.5780
 Bibcode:
 2014arXiv1404.5780D
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 14 pages