Totally Odd -subdivisions in 4-chromatic Graphs

We prove the conjecture made by Bjarne Toft in 1975 that every 4-chromatic graph contains a subdivision of in which each edge of corresponds to a path of odd length. As an auxiliary result we characterize completely the subspace of the cycle space generated by all cycles through two fixed edges. Toft's conjecture was proved independently in 1995 by Wenan Zang.