Small $2$-coloured Path Decompositions

Abstract

Consider a complete graph of multiplicity $2$, where between every pair of vertices there is one red and one blue edge. Can the edge set of such a graph be decomposed into isomorphic copies of a $2$-coloured path of length $2k$ that contains $k$ red and$k$ blue edges? A necessary condition for this to be true is $n(n-1)\equiv 0\mathrm{mod}k$. We show that this is sufficient for $k\leqq 3$.