Edge-disjoint odd cycles in planar graphs

We prove τodd(G)⩽2νodd(G) for each planar graph G where νodd(G) is the maximum number of edge-disjoint odd cycles and τodd(G) is the minimum number of edges whose removal makes G bipartite, i.e. which meet all the odd cycles. For each k, there is a 3-connected planar graph Gk with τodd(Gk)=2k and νodd(Gk)=k.