On the definability of properties of finite graphs

AbstractThis paper considers the definability of graph-properties by restricted second-order and first-order sentences. For example, it is shown that the class of Hamiltonian graphs cannot be defined by monadic second-order sentences (i.e., if quantification over the subsets of vertices is allowed); any first-order sentence that defines Hamiltonian graphs on n vertices must contain at least 12n quantifiers. The proofs use Fraïssé-Ehrenfeucht games and ultraproducts