On the bipartite density of regular graphs with large girth

Let B(G) be the edge set of a bipartite subgraph of a graph G with the maximum number of edges. Let b_k = inf{|B(G)|/|E(G)‖G is a cubic graph with girth at least k}. We will prove that lim_{k → ∞} b_k ≥ 6/7.