16 research outputs found
A new sufficient condition for a Digraph to be Hamiltonian-A proof of Manoussakis Conjecture
Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following
conjecture.
\noindent\textbf{Conjecture}. {\it Let be a 2-strongly connected digraph
of order such that for all distinct pairs of non-adjacent vertices ,
and , , we have . Then is Hamiltonian.}
In this paper, we confirm this conjecture. Moreover, we prove that if a
digraph satisfies the conditions of this conjecture and has a pair of
non-adjacent vertices such that , then
contains cycles of all lengths .Comment: 24 page
A Note on Long non-Hamiltonian Cycles in One Class of Digraphs
Let be a strong digraph on vertices. In [3, Discrete Applied
Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the
following theorem: if (*) and for every pair of non-adjacent vertices
with a common in-neighbour or a common out-neighbour, then is hamiltonian.
In this note we show that: if is not directed cycle and satisfies the
condition (*), then contains a cycle of length or .Comment: 7 pages. arXiv admin note: substantial text overlap with
arXiv:1207.564
A sufficient condition for a balanced bipartite digraph to be hamiltonian
We describe a new type of sufficient condition for a balanced bipartite
digraph to be hamiltonian. Let be a balanced bipartite digraph and be
distinct vertices in . dominates a vertex if
and ; in this case, we call the pair dominating. In
this paper, we prove that a strong balanced bipartite digraph on
vertices contains a hamiltonian cycle if, for every dominating pair of vertices
, either and or and
. The lower bound in the result is sharp.Comment: 12 pages, 3 figure