[Paper Review] On pre-Hamiltonian Cycles in Hamiltonian Digraphs

Samvel Kh. Darbinyan|arXiv (Cornell University)|Apr 30, 2014

graph theory and CDMA systems6 references3 citations

TL;DR

This paper proves that every strongly connected digraph of order $ n \geq 4 $ satisfying Manoussakis' $ A_0 $ condition either contains a pre-Hamiltonian cycle (a cycle of length $ n-1 $) or is isomorphic to the complete bipartite digraph $ K^*_{n/2,n/2} $ when $ n $ is even. The result strengthens Manoussakis' Hamiltonian cycle theorem by establishing the existence of a long cycle as a necessary intermediate structure, with tight degree bounds shown to be optimal via extremal constructions.

ABSTRACT

Let $D$ be a strongly connected directed graph of order $n\geq 4$. In \cite{[14]} (J. of 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 $x,y,z$ of vertices such that $x$ and $y$ are non-adjacent: If there is no arc from $x$ to $z$, then $d(x)+d(y)+d^+(x)+d^-(z)\geq 3n-2$. If there is no arc from $z$ to $x$, then $d(x)+d(y)+d^-(x)+d^+(z)\geq 3n-2$. Then $D$ is Hamiltonian. In this paper we show that: If $D$ satisfies the condition of Manoussakis' theorem, then $D$ contains a pre-Hamiltonian cycle (i.e., a cycle of length $n-1$) or $n$ is even and $D$ is isomorphic to the complete bipartite digraph with partite sets of cardinalities $n/2$ and $n/2$.

Motivation & Objective

• To characterize the structure of strongly connected digraphs satisfying Manoussakis' $ A_0 $ condition beyond just Hamiltonicity.
• To establish the existence of a pre-Hamiltonian cycle (cycle of length $ n-1 $) as a structural intermediate property in such digraphs.
• To identify the extremal case where no pre-Hamiltonian cycle exists, namely when $ n $ is even and $ D $ is isomorphic to $ K^*_{n/2,n/2} $.
• To show that the degree bound $ 3n-2 $ in the $ A_0 $ condition is best possible through explicit extremal examples.

Proposed method

• Adapts key claims from the proof of Manoussakis' $ A_0 $ theorem to analyze cycle structures in digraphs satisfying the condition.
• Employs a case analysis on vertex adjacency and neighborhood structures, particularly focusing on pairs $ (y_1, y_2) $ and $ (x_1, x_2) $, to derive degree constraints.
• Uses Lemma 3.2 to bound the number of edges a vertex can have within a cycle subpath, leading to contradictions when degree conditions are violated.
• Applies strong connectivity and non-adjacency assumptions to derive neighborhood equalities, such as $ N^+(y_1) = \{y_2, x_2, x_4, \dots, x_{n-2}\} $, under specific configurations.
• Constructs a cycle of length $ n-2 $ to derive contradictions when a pre-Hamiltonian cycle is assumed not to exist, forcing the digraph into the bipartite structure.
• Uses symmetry and iterative reasoning on vertex indices to show that $ x_i x_{i-1} \in D $ and $ x_i x_j \notin D $ for non-consecutive $ i,j $, leading to independent sets of size $ n/2 $.

Experimental results

Research questions

• RQ1 Does every digraph satisfying Manoussakis' $ A_0 $ condition contain a pre-Hamiltonian cycle?
• RQ2 What structural exceptions exist when no pre-Hamiltonian cycle exists?
• RQ3 Is the degree bound $ 3n-2 $ in the $ A_0 $ condition tight?
• RQ4 Can the existence of a pre-Hamiltonian cycle be used to infer pancyclicity or Hamiltonian bypasses?

Key findings

• Any strongly connected digraph of order $ n \geq 4 $ satisfying the $ A_0 $ condition contains a pre-Hamiltonian cycle or is isomorphic to $ K^*_{n/2,n/2} $ when $ n $ is even.
• The bound $ 3n-2 $ in the $ A_0 $ condition is best possible, as shown by extremal examples: the disjoint union of two complete digraphs sharing one vertex and the complete bipartite digraph minus one arc.
• If $ D $ is not isomorphic to $ K^*_{n/2,n/2} $, then $ D $ must contain a cycle of length $ n-1 $, which is a key structural feature.
• The proof shows that under the $ A_0 $ condition, the vertex set can be partitioned into two independent sets of size $ n/2 $, each forming a partite set of $ K^*_{n/2,n/2} $, when no pre-Hamiltonian cycle exists.
• The existence of a pre-Hamiltonian cycle is guaranteed unless the digraph is isomorphic to $ K^*_{n/2,n/2} $, which is the only extremal case.
• The result implies that $ D $ contains a Hamiltonian bypass (a Hamiltonian cycle with one arc reversed) unless $ D $ is isomorphic to $ K^*_{n/2,n/2} $, with the exception of the 5-vertex tournament.