Quadratic-error edge bound for representing digraphs

Establish that there exists an absolute constant c>0 such that, for every 2-weakly compatible split system S on an n-element set and every associated digraph G_r(S)=(V,E), the inequality #E≤2#V+cn^2 holds.

Background

The paper studies the maximum size M_2(n) of a 2-weakly compatible split system on n taxa. Each split system is represented by a directed graph G_r(S), rooted at a taxon r, whose vertices correspond to the parts of splits not containing r and whose edges encode the addition of a single taxon; these edges correspond to x-pairs of the split system.

Theorem 2 establishes the general bound #E≤(5/2)#V+cn^2 for the representing digraph. The authors note that this coefficient is asymptotically sharp for their current graph method, but conjecture that a stronger bound with coefficient 2 and the same quadratic error term should hold. They state that this conjectured estimate would imply M_2(n)=O(n^2 log n), substantially improving the paper’s O(n^{5/2}) result.