Vlasev–Yeats conjecture on free variables in quadratic spanning forest identities

Let $m$ be the number of marked vertices in a graph. A quadratic spanning forest identity of the indicated type has coefficients as free variables, where the left-hand side consists of a spanning tree and a spanning forest with each marked vertex in a different tree, and the right-hand side consists of pairs of spanning forests partitioning the marked vertices between two and between $m-1$ trees. Vlasev–Yeats conjecture. The formulae for quadratic spanning forest identities of this type on $m$ marked vertices have $m(m-2)$ free variables. This conjecture generalizes the known 4-vertex identity, which has eight free variables; the paper subsequently gives a formal vector-space formulation and verifies the count in the cases discussed, while the general assertion is presented as a conjecture.