Maximum Sum of Bandwidths

The bandwidth of a graph is the minimum, over vertex labelings using distinct integers, of the maximum diierence between labels of adjacent ver-tices. Let f(n) denote the maximum sum of the bandwidths of complementary n-vertex graphs. Chinn, Chung, Erd} os, and Graham 3] proved the existence of constants c 1 ; c 2 such that 2n ? c 2 log n f(n) 2n ? c 1 log n. We sharpen this to 2n ? 8 log 2 n ? O(1) f(n) 2n ? 4 log 2 n + o(log n). The proofs amount to improving the bounds on the Ramsey and Turr an numbers of the \halfgraph". When the vertices of a graph are labeled injectively with integers, the dilation of an edge is the diierence between the labels on its endpoints. The bandwidth B(G) of the graph G is the minimum, over all such labelings, of the maximum edge dilation. Chinn, Chung, Erd} os, and Graham 3] investigated the sum B(G) + B(G), where G denotes the complement of G. They proved that B(G) + B(G) n ? 2 whenever G has n vertices (for n 4). Equality holds when G is a 4-vertex path. They also established the existence of constants c 1 ; c 2 such that B(G) + B(G) < 2n ? c 1 log n for every n-vertex G and B(G) + B(G) > 2n ? c 2 log n for almost every n-vertex G. We study these constants, proving that THEOREM 1. If f(n) is the maximum of B(G) + B(G) over n-vertex graphs, then 2n ? 8 log 2 n ? O(1) f(n) 2n ? 4 log 2 n + o(log n):