Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph G with girth at least 5 is packable.The conjecture was proved by Faudree et al. with the additional condition that G has at most 65n−2 edges. In this paper, for each integer k≥3, we prove that every non-star graph with girth at least 5 and at most 2k−1kn−αk(n) edges is packable, where αk(n) is o(n) for every k. This implies that the conjecture is true for sufficiently large planar graphs.