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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 to a survey (published in J.Comb.Th.(A)).

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).

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 to a survey (published in J.Comb.Th.(A)).

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).

PS. See also Strongly regular decompositions of the complete graph.

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 to a survey (published in J.Comb.Th.(A)).

Specifically, if you further assume (not needed if $c=2$) that the adjacency matrices of these coloured graphs commute then you are quite close to the setting of amorphic association schemes; specifically for $c=3$ you need just one pair of these graphs to commute, to get an amorphic association scheme (not sure how far this can be pushed).

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 to a survey (published in J.Comb.Th.(A)).

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).

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 to a survey (published in J.Comb.Th.(A)).

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 to a survey (published in J.Comb.Th.(A)).

Specifically, if you further assume (not needed if $c=2$) that the adjacency matrices of these coloured graphs commute then you are quite close to the setting of amorphic association schemes; specifically for $c=3$ you need just one pair of these graphs to commute, to get an amorphic association scheme (not sure how far this can be pushed).