On judicious bisections of graphs

A bisection of a graph G is a bipartition S1, S2 of V (G) such that −1 ≤ |S1|− |S2| ≤ 1. It is NP-hard to find a bisection S1, S2 of a graph G maximizing e(S1, S2) (respectively, minimizing max{e(S1), e(S2)}), where e(S1, S2) denotes the number of edges of G between S1 and S2, and e(Si) denotes the number of edges of G with both ends in Si. There has been algorithmic work on bisections, but very few extremal results are known. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a bisection S1, S2 such that max{e(S1), e(S2)} ≤ m/3. In this paper, we confirm this conjecture and show that the triangle is the only extremal graph. Moreover, the bound m/3 cannot be improved to (1/3 − )m, for any > 0, by excluding K3 or by increasing the minimum degree from 2 to 3.