Bipartite subgraphs of graphs with maximum degree three

We prove that for a connected graph G with maximum degree 3 there exists a bipartite subgraph of G containing almost 3/4 of the edges of G. Furthermore, we completely characterize the set of all extremal graphs, i.e. all connected graphs G = (V, E) with maximum degree 3 for which no bipartite subgraph has more than 3 \E\ - 1/4 of the edges; \E\ denotes the cardinality of E. For 2-edge-connected graphs there are two kinds of extremal graphs which realize the lower bound 3/4 \E\.