On edge connectivity and parity factor

By Petersen's Theorem, a bridgeless cubic graph has a 2-factor. Fleischner (Discrete Math., 101, 33-37 (1992)) has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let m_e > 0 be even and m_o > 0 be odd. In this paper, we prove that every m_e-edge-connected graph with minimum degree at least m_e + 1 contains an even factor with minimum degree at least m_e and every (m_o + 1)- edge-connected graph contains an odd factor with minimum degree at least m_o, which further extends Fleischner's result. Moreover, we show that our results are best possible.