I'm trying to prove that there is a way to color (edges) a $K_{2t}$ graph with $t$ colors so it won't contain any monochromatic cycles.
I thought to display the graph as a disjoint union of $t$ paths ($P_{2t-1}$) (each path in different color, no same edges in each of the paths).
This work great in theory, the only thing I'm not sure of is if I can really create $t$ paths ($P_{2t-1}$). It sounds and seems good but how can I prove its right?