On k-resonance of grid graphs on the plane, torus and cylinder

Grid graphs on the plane, torus and cylinder are finite 2-connected bipartite graphs embedded on the plane, torus and cylinder, respectively, whose every interior face is bounded by a quadrangle. Let k be a positive integer, a grid graph is k -resonant if the deletion of any i ≤ k vertex-disjoint quadrangles from G results in a graph either having a perfect matching or being empty. If G is k -resonant for any integer k ≥ 1 , then it is called maximally resonant. In this study, we provide a complete characterization for the k -resonance of grid graphs P m × P n on plane, C m × C n on torus and P m × C n on cylinder.