A vizing-type theorem for matching forests

A well known Theorem of Vizing states that one can colour the edges of a graph by $\Delta +\alpha$ colours, such that edges of the same colour form a matching. Here, $\Delta$ denotes the maximum degree of a vertex, and $\alpha$ the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by $\Delta +\alpha$ colours, such that arcs of the same colour form a branching. For a digraph, $\Delta$ denotes the maximum indegree of a vertex, and $\alpha$ the maximum multiplicity of an arc

Reconstructing a phylogenetic level-1 network from quartets

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on n taxa from the set of all quartets containing a certain fixed taxon, in O(n^3) time. We also present a more general method which can handle more diverse quartet data, but which takes O(n^6) time. Both methods proceed by solving a certain system of linear equations over GF(2). For a general dense quartet set (containing at least one quartet on every four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an (O(n^2) sized) certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set