Nontrivial independent sets of bipartite graphs and cross-intersecting families

Let G(X,Y) be a connected, non-complete bipartite graph with |X|⩽|Y|. An independent set A of G(X,Y) is said to be trivial if A⊆X or A⊆Y. Otherwise, A is nontrivial. By α(X,Y) we denote the maximum size of nontrivial independent sets of G(X,Y). We prove that if the automorphism group of G(X,Y) is transitive and primitive on X and Y, respectively, then α(X,Y)=|Y|−d(X)+1, where d(X) is the degree of vertices in X. We also give the structures of maximum-sized nontrivial independent sets of G(X,Y). Consequently, these results give the sizes and structures of maximum-sized cross-t-intersecting families of finite sets, finite vector spaces and permutations, as well as the sizes and structures of maximum-sized cross-Sperner families of finite sets and finite vector spaces.