It is known that the orientable genus of union of two (disjoint) graphs is the sum of their genus. So, it is natural to ask


*What can be said about the non-orientable genus of union of two (disjoint) graphs?* 


Note that upper and lower bounds are known for the non-orientable genus of $k$-amalgams of two graphs with $k\geq2$ as well as the precise value of the non-orientable genus for $k=1$.



Non-orientable genus of union of graphs
The non-orientable genus $\tilde{\rm{g}}$ is not additive: consider the union of $K\_5$ and $K\_7$. We have $\tilde{\rm{g}}(K\_5)=1$, $\tilde{\rm{g}}(K\_7)=3$ and $\tilde{\rm{g}}(K\_5\cup K\_7)=3$: embed $K\_5$ on the projective plane, $K\_7$ on the torus, and take the connected sum of these two surfaces.


