Judicious Partitions of Hypergraphs

Judicious Partitions of Hypergraphs

We prove the asymptotically best possible result that, for every integer k ≥ 2, every 3-uniform graph with m edges has a vertex-partition into k sets such that each set contains at most (1+o(1))m/k edges. We also consider related problems and conjecture a more general result.

Judicious partitions of uniform hypergraphs

The vertices of any graph with m edges can be partitioned into two parts so that each part meets at least 2m 3 edges. Bollobás and Thomason conjectured that the vertices of any r-uniform graph may be likewise partitioned into r classes such that each part meets at least cm edges, with c = r 2r−1 . In this paper, we prove this conjecture for the case r = 3. In the course of the proof we shall al...

Judicious Partitions of Graphs and Hypergraphs

Judicious Partitions of 3-uniform Hypergraphs

A conjecture of Bollobás and Thomason asserts that, for r ≥ 1, every r-uniform hypergraph with m edges can be partitioned into r classes such that every class meets at least rm/(2r−1) edges. Bollobás, Reed and Thomason [3] proved that there is a partition in which every edge meets at least (1 − 1/e)m/3 ≈ 0.21m edges. Our main aim is to improve this result for r = 3. We prove that every 3-unifor...

Judicious partitions of graphs

The problem of finding good lower bounds on the size of the largest bipartite subgraph of a given graph has received a fair amount of attention. In particular, improving a result of Erdős ([10]; see also [11] for related problems), Edwards [9] proved the essentially best possible assertion that every graph with n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges....