Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $c k$ regular graph on $v$ vertices that can be given an edge coloring with $c$ colors, such that the edges corresponding to each color form a $(v,k,\lambda,\mu)$ strongly regular graph? 


For what parameters is the $c$-edge-colored $c k$ regular graph unique up to isomorphism? 


Are there any known results, is there any literature on this topic? 


This question is related to [the questions](http://maths-people.anu.edu.au/~leopardi/Leopardi-ADTHM-graph-question.pdf) asked recently at [a meeting in Banff](https://www.birs.ca/events/2014/2-day-workshops/14w2199), for the case $c=2$.



Regular graphs with strongly regular edge colorings
Such objects are closely related to *amorphic association schemes*; these are association schemes for which any merging of classes is again an association scheme.Such an object often leads to an object in the question (and there are quite often many non-isomorphic examples). 
Here is a [link](https://pure.uvt.nl/portal/files/996353/2008-56.pdf) to a survey
(published in [J.Comb.Th.(A)](http://dx.doi.org/10.1016/j.jcta.2009.03.018)).


Moreover, for $c\leq 3$ one always gets an amorphic association scheme. For $c=2$ this is trivial, and for $n=3$ this is discussed in [Sect. 7 of J.Comb.Th.(A)](http://dx.doi.org/10.1016/j.jcta.2009.03.018). 


PS. See also [Strongly regular decompositions of the complete graph](https://pure.uvt.nl/portal/files/511356/Strongly.pdf).


