Degrees of generators of phylogenetic semigroups on graphs

The Jukes Cantor binary model associates to a trivalent tree a projective toric variety, or equivalently a lattice polytope. This phylogenetic model has been studied by Sturmfels, Sullivant, Buczyńska, Wiśniewski, Xu. Buczyńska generalized this construction to trivalent graphs, in which case a graded affine semigroup is associated to a trivalent graph.

In this talk we discuss how the first Betti number of a trivalent graph is related to the degree of minimal generation of the associated semigroup. Specifically, the semigroup of a trivalent graph with the first Betti number equal to $g$ is minimally generated in degree less or equal to $g+1$. Furthermore, there are many graphs for which the bound is effective. The caterpillar graph with $g$ cycles is generated in degree $g+1$, for $g$ even, and in degree $g$, for $g$ odd.

This talk is based on the joint work with Weronika Buczyńska, Jarosław Buczyński, Mateusz Michalek.