A note on balanced bipartitions

Abstract

A balanced bipartition of a graph G is a bipartition V1 and V2 of V(G) such that -1≤|V1|-|V2|≤1. Bollobs and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V1,V 2 such that maxe(V1),e(V2)≤m3, where e(Vi) denotes the number of edges of G with both ends in Vi. In this note, we prove this conjecture for graphs with average degree at least 6 or with minimum degree at least 5. Moreover, we show that if G is a graph with m edges and n vertices, and if the maximum degree Δ(G)=o(n) or the minimum degree δ(G)→∞, then G admits a balanced bipartition V 1,V2 such that maxe(V1),e(V2) ≤(1+o(1))m4, answering a question of Bollobs and Scott in the affirmative. We also provide a sharp lower bound on maxe(V1,V2):V 1,V2 is a balanced bipartition of G, in terms of size of a maximum matching, where e(V1,V2) denotes the number of edges between V1 and V2. © 2010 Elsevier B.V.