The Maximum Genus of Graph Bundles

The upper embeddability of Cartesian and strong graph bundles with non-trivial base and fibre is proved. A similar result is obtained for both versions of lexicographic bundles. As a corollary the upper embeddability of Cartesian, strong and lexicographic products is obtained. The results cannot be generalized to the Cartesian and strong bundles with discrete fibres, i.e., to covering graphs. In this case sharp upper and lower bounds for Betti defficiency are obtained