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A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A recent paper [M.S. Rahman, M. Kaykobad, On Hamiltonian cycles and Hamiltonian paths, Information Processing Letters 94 (2005) 37-41] presents distance based sufficient conditions for the existence of a Hamiltonian path. In this paper we establish that the same condition forces Hamiltonian cycle to be present excepting for the case where end points of a Hamiltonian path is at a distance of 2.

We investigate the possibility of spectrally efficient 100 Gb/s transmission using IEEE 802.11a OFDM PHY based coherent optical OFDM and confirm that such systems may operate at 115.2 Gb/s with 26.5 GHz signal bandwidth.

In this paper, we theoretically analyze and demonstrate that spectral efficiency of a conventional direct detection based optical OFDM system (DDO-OFDM) can be improved significantly using frequency interleaving of adjacent DDO-OFDM channels where OFDM signal band of one channel occupies the spectral gap of other channel and vice versa. We show that, at optimum operating condition, the proposed technique can effectively improve the spectral efficiency of the conventional DDO-OFDM system as much as 50%. We also show that such a frequency interleaved DDO-OFDM system, with a bit rate of 48 Gb/s within 25 GHz bandwidth, achieves sufficient power budget after transmission over 25 km single mode fiber to be used in next-generation time-division-multiplexed passive optical networks (TDM-PON). Moreover, by applying 64- quadrature amplitude modulation (QAM), the system can be further scaled up to 96 Gb/s with a power budget sufficient for 1:16 split TDM-PON.