#### Title:

On pre-Hamiltonian Cycles in Hamiltonian Digraphs

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#### Uncontrolled Keywords:

Digraphs ; cycles ; Hamiltonian cycles ; pre-Hamiltoniancycles ; longest non-Hamiltonian cycles

#### Abstract:

Let D be a strongly connected directed graph o f order n > 4. In [14] (J. o f Graph Theory, Vol.16, No. 5, 51-59, 1992) Y. Manoussakis proved the following theorem: Suppose that D satisfies the following condition for every triple x ,y ,z o f vertices such that x and y are nonadjacent: If there is no arc from x to z, then d(x) + d (y) + d+ (x) + d- (z) > 3n — 2. If there is no arc from z to x, then d(x) + d(y) + d- (x) + d+(z) > 3n — 2. Then D is Hamiltonian. In this paper we show that: If D satisfies the condition o f Manoussakis’ theorem, then D contains a pre-Hamiltonian cycle (i.e., a cycle o f length n — 1) or n is even and D is isomorphic to the complete bipartite digraph with partite sets o f cardinalities n /2 and n /2 .

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#### Additional Information:

samdarbin@ipia.sci.am ; isko@ipia.sci.am

#### Affiliation:

Institute for Informatics and Automation Problems

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#### Conference title:

10 th International Conference on Computer Science and Information Technologies CSIT 2015