Let $G=(V,E)$ be a finite simple $k-$regular graph ($k\geq 1$). Does $G$ necessarily contain a subset $E'\subset E$ of edges such that only isolated edges and cycles occur as connected components in $(V,E')$?
(The answer is easily yes for $k=1,2$.)
A counterexample would easily give a counterexample to question "Antipodal" maps on regular graphs? in the case $D=2$ by considering the complementary graph of $G$ (respectively of two disjoint copies of $G$ if $G$ is "too small").