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Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. (English) Zbl 0809.05055
Let $$G$$ be a free group. The reduced rank of $$G$$ is defined as $$\widetilde r(G)= \max\{\text{rank}(G)- 1,0\}.$$ W. D. Neumann [Lect. Notes Math. 1456, 161-170 (1990; Zbl 0722.20016)] asked if the following statement is true: If $$H$$ and $$K$$ are subgroups of $$G$$ then $\sum_{HgK\in H\backslash G/K}\widetilde r(H^ g\cap K)\leq \widetilde r(H) \widetilde r(K).$ This statement is called the strengthened Hanna Neumann conjecture because it is a stronger version of a still unsolved conjecture made by Hanna Neumann [Publ. Math. 4, 186-189 (1956; Zbl 0070.02001)]. Here $$H^ g$$ denotes $$g^{-1} Hg$$, and $$H\backslash G/K$$ denotes the set of double cosets $$HgK$$, $$g\in G$$.
In this article it is shown that this new conjecture is equivalent to the following previously uncontemplated conjecture, called the amalgamated graph conjecture: If $$\Delta$$ is any finite bipartite graph for which there exist three finite bipartite $$\Delta$$-graphs $$A$$, $$B$$, $$\Gamma$$ such that the finite bipartite graph $$(A\vee_ \Delta B)\vee (B\vee_ \Delta \Gamma)\vee (\Gamma\vee_ \Delta A)$$ is simple-edged and is the disjoint union of two isomorphic bipartite subgraphs, then $$\Delta$$ has at most half as many edges as the complete bipartite graph with the same vertex set. Terminology used here is as follows: a bipartite graph has each vertex coloured black or white, and each edge joins two vertices of different colours; a $$\Delta$$-graph is a graph given with a specified copy of $$\Delta$$ as a bipartite subgraph; simple-edged means two vertices are joined by at most one edge, $$\vee$$ denotes disjoint union, and $$\vee_ \Delta$$ the pushout identifying the two copies of $$\Delta$$. Simple proofs of some previously known results are given to demonstrate the strength of this equivalence.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20E05 Free nonabelian groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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##### References:
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