Simple probabilistic analysis to generalize bottleneck graph multi-partitioning

What is the smallest Φ ( h , k , m ) such that for any graph G = ( V , E ) involving m edges and any integer k ≥ 2 h for h ≥ 1 , there is a partition V = ∪ i = 1 k V i such that the number of edges induced by the union of any h parts is at most Φ ( h , k , m ) ? For h = 1 and 2, this coincides with the judicious partitioning problems proposed by Porter (1992) in [1] and by Bollobás and Scott in [B. Bollobás, A. D. Scott, Problems and results on judicious partitions, Random Structure Algorithms, 21 (2002), 414–430]. We show that ( h − 1 ) m k − 1 ≤ Φ ( h , k , m ) ≤ ( h − 1 2 h − 2 ) m k + O ( m 4 5 ) for general k ≥ 2 h for h ≥ 2 , and for certain cases Φ ( 2 , k , m ) ≤ 1.5 m / k + O ( m 4 5 ) improves on previous results for h = 2 .