Intersection of two deterministic parity automata Given two deterministic parity automata $A_1=(Q_1,\Sigma,\delta,q_{01},c_1)$ and $A_2=(Q_2,\Sigma,\delta,q_{02},c_2)$ with the finite set of states $Q_i$, the finite alphabet $\Sigma_i$, the transition function $\delta_i : Q_i \times \Sigma \rightarrow Q_i$, the initial state $q_{0i}$ in $Q_i$, and the coloring function $c_i : Q_i \rightarrow \mathbb{N}$, with $i \in\{0,1\}$.

How is the intersection of two deterministic parity automata formally defined? In particular, how are the coloring functions combined?
The following paper:
http://www.faculty.idc.ac.il/udiboker/files/AutomataTypes.pdf
shows that the intersection (and union) of deterministic parity automata may involve an exponential blowup. 

Thus, you cannot do much better than converting to nondeterministic Buchi automata, and taking the intersection there (and determinizing back to parity).