Graphs, intersections of subgroups of free groups and corank

The topic of this paper is the discussion of a combinatorial problem on certain edge-labeled graphs, motivated by a conjecture in the theory of free groups. oup-theoretic question is the following. It is known that every subgroup of a free group F(A) is free, and that the intersection of finitely generated subgroups is finitely generated (Howson, 1954 [4]). Hanna Neumann proved the following inequality, ∩K) −1 ≤2(rank (H) −1)(rank(K −1), njectured (1956 [7]) that in fact, ∩K) − 1 ≤(rank(H) −1)(rank(K)−1). ediate results appeared in the literature since, proving in particular that the conjecture holds when H or K has rank at most 3 (see below), but the general case is still open. in result which we discuss here is the proof that the conjecture holds when one of the subgroups is positively generated, that is, it admits a system of generators which can be written without using the inverses of the letters in A. This result is due to 3. Meakin and the author and a complete proof can be found in [6]. We also refer to [6] for a more thorough bibliography. Another proof was found independently by B. Khan [3]. st results known to date on H. Neumannʹs conjecture are briefly recalled in the first section. Then we make explicit the classical connection between the subgroups of free groups and labeled directed graphs and we discuss a graph-theoretic statement whose proof would imply H. Neumannʹs conjecture. In the third section we summarize the main arguments in the proof a particular case of the conjecture and its application to the positively generated case. We conclude with an open discussion on the exact scope of the partial result we obtained.