Optimal linear arrangement of a rectangular grid

Abstract

An optimal linear arrangement of a finite simple graph G=( V, E) with vertex set V, edge set E, and | V|= N, is a map f from V onto {1,2,…, N} that minimizes ∑ { u, v}∈ E | f( u)− f( v)|. We determine optimal linear arrangements for m× n rectangular grids where V={1,2,…, m}×{1,2,…, n} and E={{(i,j),(k,ℓ)} : |i−k|+|j−ℓ|=1} . When m⩾ n⩾5, they are disjoint from bandwidth-minimizing arrangements for which f minimizes the maximum | f( u)− f( v)| over E. The different solutions to the bandwidth and linear arrangement problems for rectangular grids is reminiscent of Harper's result (J. Soc. Ind. Appl. Math. 12 (1964) 131–135; J. Combin. Theory 1 (1966) 385–393) of different bandwidth and linear arrangement solutions for the hypercube graph with vertex set {0,1} n and edge set {{(x 1,x 2,…,x n), (y 1,y 2,…,y n)} : ∑ i|x i−y i|=1} .