Triangle-free subcubic graphs with minimum bipartite density

A graph is subcubic if its maximum degree is at most 3. The bipartite density of a graph G is max{ε(H)/ε(G) : H is a bipartite subgraph of G}, where ε(H) and ε(G) denote the numbers of edges in H and G, respectively. It is an NP-hard problem to determine the bipartite density of any given triangle-free cubic graph. Bondy and Locke gave a polynomial time algorithm which, given a triangle-free subcubic graph G, finds a bipartite subgraph of G with at least 4 5 ε(G) edges; and showed that the Petersen graph and the dodecahedron are the only triangle-free cubic graphs with bipartite density 4 5 . Bondy and Locke further conjectured that there are precisely seven triangle-free subcubic graphs with bipartite density 4 5 . We prove this conjecture of Bondy and Locke. Our result will be used in a forthcoming paper to solve a problem of Bollobás and Scott related to judicious partitions.