Given that a 3regular multigraph is 3edgecolorable, is there an expression for how many 3edgecolorings exist?
(For example, if a 2regular multigraph is 2edgecolorable, there are $2^k$ 2edgecolorings where $k$ is the number of cycles.)
Given that a 3regular multigraph is 3edgecolorable, is there an expression for how many 3edgecolorings exist? (For example, if a 2regular multigraph is 2edgecolorable, there are $2^k$ 2edgecolorings where $k$ is the number of cycles.) 

