Minimum balanced bipartitions of planar triangulations

Abstract

A balanced bipartition of a graph G is a bipartition (V1,V2) of V(G) where V1 and V2 differ in size by at most 1. A minimum balanced bipartition of G is a balanced bipartition (V1,V2) of V(G) with the minimum number e(V1,V2) of edges with ends in both V1 and V2. We show that, for every plane triangulation G, there exists a minimum balanced bipartition (V1,V2) of V(G) with e(V1,V2)≤|V(G)| such that both V1 and V2 induce connected near-triangulations, and the total number of blocks in G[V1] and G[V2] exceeds the total number of internal vertices by at most 2. This confirms the folklore conjecture that, for any planar graph G, a minimum balanced bipartition (V1,V2) of V(G) has e(V1,V2)≤|V(G)|.