$K_9$ edge colouring Assume we have $K_9$ graph and $4$ colours. We want to colour its edges and obtain following property:
If we change colours in our colouring, such graph will be isomorphic to the previous one (before colour change). Moreover, such graph is unique respect to automorphism group.
How to do it?
So far, I guessed that every vertice should be connected with 2 edges in all 4 colours (for example 2 blue, 2 green, 2 red, 2 yellow).
 A: 
If we change colours in our colouring, such graph will be isomorphic to the previous one

This implies that each one of the 4 coloured subgraphs must be identical (isomorphic), so you have a decomposition of $K_9$ into 4 identical graphs.
Do you know of any such graph decomposition ?

 Additional help, as you said you need that each vertex has degree 2 in each subgraph. And each subgraph contains 9 vertices and 9 edges.

A: Here is an example of how to achieve most of what you are describing - except I don't really understand your requirement "such graph is unique respect to automorphism group." Each colour is a set of $3$ $C_3$ graphs. ($C_3$ is a triangle so it is also $K_3$).

My attempt to meet the requirement by decomposing into $C_9$ graphs ran into trouble. If this is possible I would like to see one - update:@ThomasLesgourgues was kind enough to point me at the Wikipedia for Hamiltonian Decomposition which shows how $K_9$ has such a set of spanning $C_9$ factor graphs. and if not, it would be great to establish why putting in three such $\require{enclose}\enclose{horizontalstrike}{C_9}$ cycles results in the last being $\enclose{horizontalstrike}{3\times C_3}$.
