Let $G$ be a regular graph having spanning regular subgraphs $G\_1,\dots, G\_k$
whose edge sets are disjoint and their union is the whole edge set of $G$.
Is it true that the clique number of $G$ is bounded above by the sum of clique numbers
of $G\_i$s? If not, under what conditions the answer is positive.



Clique number of a regular graph with respect to that of a certain edge decomposition
The edges of $K\_{2n+1}$ can be split into $n$ Hamiltonian cycles, whose clique numbers are 2, which gives a counterexample to the initial conjecture. Moreover, these cycles can be merged into larger regular graphs without increasing of the clique number (check, e.g., the case when $2n+1$ is prime!), which provides much larger gap...


