Edge-bandwidth of Tensor Product of paths and cycles

The bandwidth of a graph is the minimum of the maximum difference between labels of adjacent vertices in the graph. If we label the edges instead of the vertices of the graph, we can define the edge-bandwidth accordingly. People start working on the edge-bandwidth of graphs since 1999(7). The edge-bandwidth of a graph is the minimum of the maximum difference between labels of adjacent edges in the graph. Since the edge-bandwidth of a graph G is equal to the bandwidth of the line graph of G, establishing the edge-bandwidth of a graph is equivalent to verifying the bandwidth of one or more graphs. The decision problem corresponding to find the bandwidth of an arbitrary graph is NP-complete(10). It is NP-complete even for trees of maximum degree 3(6). Although the edge-bandwidth problem is included in the bandwidth problem, the computing complexity of the edge-bandwidth is unknown up to now. The application about the edge-bandwidth is in the area of network circuit on-line routing and admission control problem. The edge-bandwidth problem has been solved for only a few classes of graphs such as complete graph, complete bipartite graph with equal partites, caterpillar and theta graph(5). This paper establishes the edge-bandwidth of the tensor product of a path with a path and a path with a cycle. Optimal edge-numberings to achieve each of these edge-bandwidths are provided.