Simple probabilistic analysis to generalize bottleneck graph multi-partitioning

What is the smallest Φ(h,k,m)Φ(h,k,m) such that for any graph G=(V,E)G=(V,E) involving mm edges and any integer k≥2hk≥2h for h≥1h≥1, there is a partition V=∪i=1kVi such that the number of edges induced by the union of any hh parts is at most Φ(h,k,m)Φ(h,k,m)? For h=1h=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)mk−1≤Φ(h,k,m)≤(h−12h−2)mk+O(m45) for general k≥2hk≥2h for h≥2h≥2, and for certain cases Φ(2,k,m)≤1.5m/k+O(m45) improves on previous results for h=2h=2.