On Petersen's graph theorem

In this paper we prove the following: let G be a graph with e_G edges, which is (k - 1)-edge- connected, and with all valences ≥k. Let 1≤r≤k be an integer, then G contains a spanning subgraph H, so that all valences in H are ≥r, with no more than ⌈re_G{plus 45 degree rule}k⌉ edges. The proof is based on a useful extension of Tutte's factor theorem [4,5], due to Lovász [3]. For other extensions of Petersen's theorem, see [6,7,8].