Min-max relations for odd cycles in planar graphs

Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and t(G) the minimum number of vertices whose removal makes G bipartite. We show that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G) by Fiorini, Hardy, Reed and Vetta [Math. Program. Ser. B 110 (2007), 71-91].