Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomi$ algorithm which returns a bipartite subgraph of G containing at least 5 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.

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A lower bound is established on the number of edges in a maximum kcolorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.