The graphic Merino–Welsh conjectures

Let $G$ be a graph with no bridges and no loops. Let $\tau(G)$ be its number of spanning trees, let $(G)$ be its number of acyclic orientations, and let $^*(G)$ be its number of totally cyclic orientations. Graphic Merino–Welsh conjectures. The following inequalities hold:

$$\max\left((G),^*(G)\right)\geq\tau(G),$$ $$(G)+^*(G)\geq 2\cdot\tau(G),$$

and

$$(G)\cdot^*(G)\geq\tau(G)^2.$$

The third, multiplicative, inequality is the strongest and implies the additive inequality, which in turn implies the first. The conjectures have been proved for several graph families, including wheels, complete graphs, and series-parallel graphs, but remain open in general.