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. njecture was proved by Faudree et al. with the additional condition that G has at most 6 5 n − 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 2 k − 1 k n − α k ( n ) edges is packable, where α k ( n ) is o ( n ) for every k . This implies that the conjecture is true for sufficiently large planar graphs.