Existence of a Trade-Off Factorization of a Partially $4$-Homogeneous Graph

We consider a graph $G$ with $2\kappa$ vertices of degree $5$ and $\kappa$ vertices of degree $2$, all other vertices being of degree $4$. In connection with the timetable optimization problem, we study necessary and sufficient conditions for the existence of a factorization of $G$ into two skeleton subgraphs whose edge sets are disjoint and have the same cardinality and, for each vertex of the graph, the numbers of edges incident to this vertex in these subgraphs differ at most by unity.