The maximum forcing numbers of quadriculated tori

Abstract

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 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 2mX2n quadriculated torus is mn. By improving Kleinerman's approach we obtain the maximum forcing numbers of all 4-regular quadriculated graphs on torus except one class.