On several partitioning problems of Bollobás and Scott

Authors:
Jie Ma;Pei-Lan Yen;Xingxing Yu
Affiliations:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA;Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2010

Citing 5
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Abstract

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let G be a hypergraph with m"i edges of size i for i=1,2. We show that for any integer k=1, V(G) admits a partition into k sets each containing at most m"1/k+m"2/k^2+o(m"2) edges, establishing a conjecture of Bollobas and Scott. We also prove that V(G) admits a partition into k=3 sets, each meeting at least m"1/k+m"2/(k-1)+o(m"2) edges, which, for large graphs, implies a conjecture of Bollobas and Scott (the conjecture is for all graphs). For k=2, we prove that V(G) admits a partition into two sets each meeting at least m"1/2+3m"2/4+o(m"2) edges, which solves a special case of a more general problem of Bollobas and Scott.