Problems and results on judicious partitions

Abstract

ABSTRACT:We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.

On Tight Components and Anti-Tight Components

“…Lee et al [6] used tight components to solve a problem of Bollobás and Scott [1,9] about the dependence on minimum degree of bounds on judicious bisections. A block in a graph G is a maximal connected subgraph that contains no cut vertex.…”

Balanced judicious bipartitions of graphs

“…Edwards [6,7] improved this lower bound to m 2 + 1 4 2m + 1 4 − 1 8 , which is essentially best possible as evidenced by the complete graphs K 2n+1 . In [4] (also see [3]), Bollobás and Scott extend Edwards' bound to k-partitions of graphs by proving that the vertex set of any graph with m edges can be partitioned into V 1 , . .…”

“…, V k ). Bollobás and Scott [3] asked an analogous question for judicious partitions: Given a graph G, find a balanced partition of V (G) into V 1 , . .…”

“…, e(V k )}. In particular, they made the following conjecture, where e(G) denotes the number of edges in the graph G. Conjecture 1.1 (Bollobás and Scott [3]) Let G be a graph with minimum degree at least 2. Then V (G) admits a balanced partition V 1 , V 2 such that e(V i ) ≤ e(G)/3 for i ∈ {1, 2}.…”

“…Bollobás and Scott [3] conjectured that if G is a graph with minimum degree at least 2 then V (G) admits a balanced bipartition V 1 , V 2 such that for each i, G has at most |E(G)|/3 edges with both ends in V i . The minimum degree condition is necessary, and a result of Bollobás and Scott [5] shows that this conjecture holds for regular graphs G (i.e., when ∆(G) = δ(G)).…”