The maximum forcing numbers of quadriculated tori

Klein and Randic (1985) proposed the concept of forcing number, which has an application in chemical resonance theory. Let $G$ be a graph with a perfect matching $M$. The forcing number of $M$ is the smallest cardinality of a subset of $M$ that is contained only in one perfect matching $M$. The maximum forcing number of $G$ is the maximum value of forcing numbers over all perfect matchings of $G$. Kleinerman (2006) obtained that the maximum forcing number of $2n\times 2m$ quadriculated torus is $nm$. By improving Kleinerman's approach, we obtain the maximum forcing numbers of all 4-regular quadriculated graphs on torus except one class.