Extending Matchings to \(2\)-Factors

Let \( G \) be a finite \( 4 \)-regular cyclically \( 2k \)-edge-connected simple graph for some integer \( k \geq 1 \). Let \( E(k) \) be a set of \( k \) independent edges in \( G \) and \( (E_1, E_2) \) be a partition of \( E(k) \). We consider when there exists a \( 2 \)-factor in \( G \) which excludes all edges of \( E_1 \), and includes all the edges of \( E_2 \). A complete characterization is provided.