On a conjecture that strengthens Kundu's k <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0001" wiley:location="equation/jgt23177-math-0001.png"><mrow><mrow><mi>k<

Abstract Let be a nonincreasing degree sequence with even . In 1974, Kundu showed that if is graphic, then some realization of has a ‐factor. For , Busch et al. and later Seacrest for showed that if and is graphic, then there is a realization with a ‐factor whose edges can be partitioned into a ‐factor and edge‐disjoint 1‐factors. We improve this to any . In 1978, Brualdi and then Busch et al. in 2012, conjectured that . The conjecture is still open for . However, Busch et al. showed the conjecture is true when or . We explore this conjecture by first developing new tools that generalize edge‐exchanges. With these new tools, we can drop the assumption is graphic and show that if then has a realization with edge‐disjoint 1‐factors. From this we confirm the conjecture when or when is graphic and .