A note on 1-factors in graphs

Conditions on a graphG are presented which are sufficient to guarantee thatG−Z contains a 1-factor, whereZ is a set of edges ofG of restricted cardinality. These conditions provide generalizations of several known results and, further, establish the result that ifG is anr-regular, (r−2)-edge-connected graph (r≥2) of even order andz is an integer with 0≤z≤r−1 such thatG contains fewer thanr−z edge cut sets of cardinalityr−2, thenG−Z has a 1-factor for each setZ ofz edges ofG.