On the optimal transversals of the odd cycles

Let G be a simple graph and let X be its vertex-set. A set T ⊆ X is a transversal of the odd cycles if it meets all the odd cycles of G. Let HG denote the family of the odd cycles of G (as subsets of X) which are chordless, i.e. minimal relatively to inclusion. Clearly, the minimum cardinality of a transversal T is the transversal number of the hypergraph HG, that is, with the notations of Hypergraph Theory (see [2]), min|T|=τ(HG) In this paper, we study the coefficient τ(HG); an unsolved problem is: For which graph G is this coefficient equal to the maximum number of pairwise disjoint odd cycles (the ‘König Property’)?