The incidence matrix of Cnm of a simple digraph is mapped into a incidence matrix F of a balanced bipartite undirected graph by divided C into two groups. Based on the mapping, it proves that the complexity is polynomial to determin a Hamiltonian cycle existence or not in a simple digraph with degree bound two and obtain all solution if it exists Hamiltonian cycle. It also proves P = NP with th...

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the inpu...

In this paper we show that the Inverse problems of HAMILTONIAN CYCLE and 3D-MATCHING are coNP complete. This completes the study of inverse problems of the six natural NP-complete problems from [GJ79] and answers an open question from [Ch03]. We classify the inverse complexity of the natural verifier for HAMILTONIAN CYCLE and 3D-MATCHING by showing coNP-completeness of the corresponding inverse...

2-connected outerplanar graphs have a unique minimal cycle basis with length 2|E| − |V |. They are the only Hamiltonian graphs with a cycle basis of this length.

In this paper, we prove that directed cyclic hamiltonian cycle systems of the complete symmetric digraph, K∗ n, exist if and only if n ≡ 2 (mod 4) and n 6= 2p with p prime and α ≥ 1. We also show that directed cyclic hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (Kn− I)∗, exist if and only if n ≡ 0 (mod 4).

The enhanced pyramid graph was recently proposed as an interconnection network model in parallel processing for maximizing regularity in pyramid networks. We prove that there are two edge-disjoint Hamiltonian cycles in the enhanced pyramid networks. This investigation demonstrates its superior property in edge fault tolerance. This result is optimal in the sense that the minimum degree of the g...

The butterfly graphs were originally defined as the underlying graphs of FFT networks which can perform the fast Fourier transform (FFT) very eficiently. Since butterfly graphs are regular of degree four, it can tolerate at most two edge faults in the worst case in order to establish a Hamiltonian cycle. In this paper, we show that butterfly graphs contain a fault-free Hamiltonian cycle even if...

We show for an arbitrary lp norm that the property that a random geometric graph G(n, r) contains a Hamiltonian cycle exhibits a sharp threshold at r = r(n) = √ log n αpn , where αp is the area of the unit disk in the lp norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of G(n, r) a.a.s., provided r = r(n) ≥

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X| ≥ 3 and if X...