Bounds for pairs in partitions of graphs

In this paper we study the following problem of Bollobas and Scott: What is the smallest f(k, m) such that for any integer k >= 2 and any graph G with m edges, there is a partition V(G) = boolean OR(k)(i=1) V-i such that for 1 <= i not equal j <= k, e(V-i boolean OR V-j) <= f(k, m)? We show that f(k, m) < 1.6m/k + o(m), and f(k, m) < 1.5m/k + o(m) for k >= 23. (While the graph K-1,K- n shows that f(k, m) >= m/(k - 1), which is 1.5m/k when k = 3.) We also show that f(4, m) <= m/3 + o(m) and f(5, m) <= 4m/15 + o(m), providing evidence to a conjecture of Bollobas and Scott. For dense graphs, we improve the bound to 4m/k(2) + o(m), which, for large graphs, answers in the affirmative a related question of Bollobas and Scott. (C) 2010 Elsevier B.V. All rights reserved.