Abstract Suppose G and H are bipartite graphs $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph induced between $L(v)$ $L(v')$ is matching, whenever $vv'\in E(G)$ . We show for each $\varepsilon>0$ if has maximum degree D $|L(v)| \ge (1+\varepsilon )D/\log D$ all $v\in V(G)$ , then admits an independent transversal with respect to L provided sufficiently large. This ...