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{@e(H)/@e(G):H is a bipartite subgraph of G}, where @e(H) and @e(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 45@e(G) edges; and showed that the Petersen graph and the dodecahedron are the only triangle-free cubic graphs with bipartite density 45. Bondy and Locke further conjectured that there are precisely seven triangle-free subcubic graphs with bipartite density 45. We prove this conjecture of Bondy and Locke. Our result will be used in a forthcoming paper to solve a problem of Bollobas and Scott related to judicious partitions.

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