The Global Packing Number for an Optical Network

The global packing number problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph G. We assume all communication channels are bidirectional, so that all links and paths are undirected. It follows that there are (|G| 2 ) distinct node pairs associated with G, where |G| is the number of nodes in G. A path system P of G consists of (|G| 2 ) paths, one path to connect each of the node pairs. The global packing number of a path system P , denoted by Φ(G,P), is the minimum integer k to guarantee the existence of a mapping ω : P → {1, 2, . . . , k}, such that ω(P ) 6= ω(P ′) if P and P ′ have common edge(s). The global packing number of G, denoted by Φ(G), is defined to be the minimum Φ(G,P) among all possible path systems P . If there is no wavelength conversion along any optical transmission path for any node pair in the network, the global packing number signifies the minimum number of wavelengths required to support simultaneous communication for all pairs in the network. Partially supported by the National Natural Science Foundation of China (No. 61301107, 61174060), the Hong Kong RGC Earmarked Grant CUHK414012 and the Shenzhen Knowledge Innovation Program JCYJ20130401172046453 (Y.-H. Lo, Y. Zhang and W. S. Wong), and by Ministry of Science and Technology, Taiwan under grants MOST 104-2115-M-009-009 (H.-L. Fu). Y.-H. Lo is with the School of Mathematical Sciences, Xiamen University, Xiamen 361005, China. E-mail: [email protected] Y. Zhang is with the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China. E-mail: [email protected] W. S. Wong is with the Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong. E-mail: [email protected] H.-L. Fu is with the Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan, ROC. E-mail: [email protected] September 24, 2015 DRAFT ar X iv :1 50 9. 07 02 9v 1 [ cs .D M ] 2 3 Se p 20 15