The structure of even factors in claw-free graphs

Recently, Jackson and Yoshimoto proved that every bridgeless simple graph G with δ ( G ) ≥ 3 has an even factor in which every component has order at least four, which strengthens a classical result of Petersen. In this paper, we give a strengthening of the above result and show that the above graphs have an even factor in which every component has order at least four that does not contain any given edge. We also extend the above result to the graphs with minimum degree at least three such that all bridges lie in a common path and to the bridgeless graphs that have at most two vertices of degree two respectively. Finally we use this extended result to show that every simple claw-free graph G of order n with δ ( G ) ≥ 3 has an even factor with at most max { 1 , ⌊ 2 n − 2 7 ⌋ } components. The upper bound is best possible.