Improved lower bounds on the compatibility of quartets, triplets, and multi-state characters

Abstract

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r ≥ 2, there exists an incompatible set C of ⌊r/ 2⌋·⌈r/2⌉ + 1 r-state characters such that every proper subset of C is compatible. Thus, f(r) ≥ ⌊r/2⌋·⌈r/ 2⌉ + 1 for every r ≥ 2. This improves the previous lower bound of f(r) ≥ r given by Meacham (1983), and generalizes the construction showing that f(4) ≥ 5 given by Habib and To (2011). We prove our result via a result on quartet compatibility that may be of independent interest: For every integer n ≥ 4, there exists an incompatible set Q of ⌊n-2/ 2⌋·⌈n-2/2⌉ + 1 quartets over n labels such that every proper subset of Q is compatible. We contrast this with a result on the compatibility of triplets: For every n ≥ 3, if R is an incompatible set of more than n - 1 triplets over n labels, then some proper subset of R is incompatible. We show this bound is tight by exhibiting, for every n ≥ 3, a set of n - 1 triplets over n taxa such that R is incompatible, but every proper subset of R is compatible. © 2012 Springer-Verlag.