CHARACTERIZING GRAPHIC MATROIDS BY A SYSTEM OF LINEAR EQUATIONS

Given a rank-r binary matroid we construct a system of O(r³) linear equations in O(r²) variables that has a solution over GF(2) if and only if the matroid is graphic. We prove the following result

Characterizing graphic matroids by a system of linear equations

Given a rank-$r$ binary matroid we construct a system of $O(r^3)$ linear equations in $O(r^3)$ variables that has a solution over GF$(2)$ if and only if the matroid is graphic