Edge-bandwidth of grids and tori

Authors:
Oleg Pikhurko;Jerzy Wojciechowski
Affiliations:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA;Department of Mathematics, West Virginia University, Morgantown, WV
Venue:
Theoretical Computer Science
Year:
2006

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Abstract

The edge-bandwidth of a graph G is the smallest number B' for which there is a bijective labeling of E(G) with {1,...,e(G)} such that the difference between the labels at any adjacent edges is at most B'. Here we compute the edge-bandwidth for rectangular grids: B'(Pm⊕Pn)=2 min(m,n) - 1 if max(m,n) ≥ 3, where ⊕ is the Cartesian product and Pn denotes the path on n vertices. This settles a conjecture of Calamoneri et al. [New results on edge-bandwidth, Theoret. Comput. Sci. 307 (2003) 503-513]. We also compute the edge-bandwidth of any torus (a product of two cycles) within an additive error of 5.