On domination numbers of graph bundles
Let $\gamma (G)$ be the domination number of a graph $G$. It is shown that for any $k\ge 0$ there exists a Cartesian graph bundle $B\bun F$ such that $\gamma (B\bun F)= \gamma(B) \gamma (F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality $\dom (B\buns F)\le \dom (B) \dom (F)$ for strong graph bundles. Examples of graphs $B$ and $F$ with $\dom (B\buns F)< \dom (B) \dom (F)$ are given.