Bounded fractionality of the multiflow feasibility problem for demand graph K 3 + K 3 and related

We consider the multiflow feasibility problem whose demand graph is the vertex-disjoint union of two triangles. We show that this problem has a 1/12-integral solution or no solution under the Euler condition. This solves a conjecture raised by Karzanov, and completes the classification of the demand graphs having bounded fractionality. We reduce this problem to the multiflow maximization problem whose terminal weight is the graph metric of the complete bipartite graph Kn,m, and show that it always has a 1/12-integral optimal multiflow for every inner Eulerian graph.