Largest bipartite subgraphs in triangle-free graphs with maximum degree three

Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomial algorithm which returns a bipartite subgraph of G containing at least ? of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3-regular, triangle-free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.