Maximum cuts and judicious partitions in graphs without short cycles

We consider the bipartite cut and the judicious partition problems in graphs of girth at least 4. For the bipartite cut problem we show that every graph G with nt edges, whose shortest cycle has length at least r greater than or equal to 4, has a bipartite subgraph with at least (m)(2) over bar + c(r)m(r)(r+1) over bar edges. The order of the error term in this result is shown to be optimal for r = 5 thus settling a special case of a conjecture of Erdos. (The result and its optimality for another special case, r = 4, were already known.) For judicious partitions, we prove a general result as follows: if a graph G = (V, E) with m edges has a bipartite cut of size (m)(2) over bar + delta, then there exists a partition V = V-1 boolean OR V-2 such that both parts V-1, V-2 span at most (m)(4) over bar (1 - o(1))(delta)(2) over bar + O(rootm) edges for the case delta = o(m), and at most ((1)(4) over bar - Ohm(1))m edges for delta = Ohm(m). This enables one to extend results for the bipartite cut problem to the corresponding ones for judicious partitioning. (C) 2003 Elsevier Science (USA). All rights reserved.