Growing balanced covering sets

Given a bipartite graph with bipartition (A,B) where B is equipartitioned into k blocks, can the vertices in A be picked one by so that at every step, cover roughly same number of each these blocks? We show that, if block has cardinality m, have degree, and vertex most cm neighbors c>0 small constant, then there an ordering v1,…,vn such for j∈{1,…,n}, numbers neighbor {v1,…,vj} two blocks differ 2(k−1)c⋅m. This related to well-known lemma Steinitz, partially answers unpublished question Scott Seymour.