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 $\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ r-state characters such that every proper subset of C is compatible. Thus, $f(r) \ge \lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ 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 $\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1$ 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.

This work was supported in part by the National Science Foundation under grants CCF-1017189 and DEB-0829674.