Offprints from the Theory of Applications of Graphs

The bandwidth b(G) of a graph G is defined by b(G) = min max 1A(-)-X(y)1 A e={x,y} where e ranges over all edges of G and A ranges over all 1 1 functions A :V(G) -> Z , the positive integers . In this note we show for any graph G on n vertices (with G denoting its complement), b(G) + b(G) > n 2 Furthermore, for all n > 3 there exist graphs which achieve this bound . We also prove : (i) b(G) + b(G) < 2n c 1 log n, for all graphs G on n vertices ; (ii) b(G) + b(G) > 2n c 2 log n, for almost all graphs G on n vertices . 1 . Introduction . For undefined graph theory terminology see [1] or [8] . The bandwidth b(G) of a graph C is defined to be the least integer b such that for some labelling A of the vertices of G