Summary: We study deterministic algorithms for the gossiping problem in ad hoc radio networks under the assumption that each combined message contains at most $b(n)$ single messages or bits of auxiliary information, where $b$ is an integer function and $n$ is the number of nodes in the network. We term such a restricted gossiping problem $b(n)$-gossiping. We show that $\sqrt n$-gossiping in an ad hoc radio network on $n$ nodes can be done deterministically in time $\tilde O(n^{3/2})$ which asymptotically matches the best known upper bound on the time complexity of unrestricted deterministic gossiping. Notation $\tilde O(f(n))$ stands for $O(f(n)\cdot\log^c n)$, for any constant $c>0$. Our upper bound on $\sqrt n$-gossiping is tight up to a poly-logarithmic factor and it implies similarly tight upper bounds on $b(n)$-gossiping where function $b$ is computable and $1\le b(n)\le \sqrt n$ holds. For symmetric ad hoc radio networks, we show that even $1$-gossiping can be done deterministically in time $\tilde O(n^{3/2})$. We also demonstrate that $O(n^t)$-gossiping in a symmetric ad hoc radio network on $n$ nodes can be done in time $O(n^{2-t})$. Note that the latter upper bound is $o(n^{3/2})$ when the size of a combined message is $ω(n^{1/2})$. Furthermore, by adopting known results on repeated randomized broadcasting in symmetric ad hoc radio networks, we derive a randomized protocol for 1-gossiping in these networks running in time $\tilde O(n)$ on the average. Finally, we observe that when a collision detection mechanism is available, even deterministic 1-gossiping in symmetric ad hoc radio networks can be performed in time $\tilde O(n)$.