Simultaneous separation in bounded degree trees

It follows from a classical result of Jordan that every tree with maximum degree at most r containing a vertex set labeled by [n], has a single-edge cut which separates two subsets A,B⊂[n] for which min{|A|,|B|}≥(n−1)/r. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of several trees: Given k trees with maximum degree at most r, containing a common vertex set labeled by [n], we ask for a single-edge cut in each tree which maximizes min{|A|,|B|} where A,B⊂[n] are separated by the corresponding cut at each tree. Denoting this maximum by f(r,k,n) and considering the limit f(r,k)=lim_n→∞f(r,k,n)/n (which is shown to always exist) we determine that f(r,2)=12r and determine that f(3,3)=227, which is already quite intricate. The case r=3 is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over n taxa have a split at each tree which separates two taxa sets of order at least n/6 (resp. 2n/27), and these bounds are asymptotically tight.