Bounds for pairs in judicious partitioning of graphs

In 2002, Bollobás and Scott posed the following problem: for an integer k≥2 and a graph G of m edges, what is the smallest f(k, m) such that V(G) can be partitioned into V 1,…,Vk in which e(Vi∪Vj)≤f(k,m) for all 1≤i≠j≤k, where e(Vi∪Vj) denotes the number of edges with both ends in Vi∪Vj? In this paper, we solve this problem asymptotically by showing that f(k,m)≤m/(k−1)+o(m). We also show that V(G) can be partitioned into V1,…,Vk such that e(Vi∪Vj)≤4m/k2+4Δ/k+o(m) for 1≤i≠j≤k, where Δ denotes the maximum degree of G. This confirms a conjecture of Bollobás and Scott. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 59–70, 2017