Long Local Searches for Maximal Bipartite Subgraphs

Given a partition of the vertices of a graph into two sets, a flip is a move of a vertex from its own set to the other, under the condition that it has more incident edges to vertices in its own set than in the other. Every sequence of flips eventually produces a bipartite subgraph capturing more than half of the edges in the graph. Each flip gains at least one edge. For an n-vertex loopless multigraph, we show that there is always a sequence of at most n/2 flips that cannot be extended, and we construct a graph having a sequence of 2 25 (n + n − 31) flips.