The Hanna Neumann Conjecture is true when one subgroup has a positive generating set

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The Hanna Neumann conjecture states that if F is a free group, then for all finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [rank(H)-1] [rank(K)-1] In this paper, we show that if one of the subgroups, say H, has a generating set consisting of only positive words, then H is not part of any counterexample to the conjecture. We further show that if H <= F(a,b) is part of a counterexample to the conjecture, then its folding Gammaₕ must contain source and/or sink vertices.