  We prove best-possible edge density conditions sufficient to guarantee
traceability, Hamiltonicity, chorded pancyclicity, Hamiltonian-connectedness,
$k$-path Hamiltonicity, and $k$-Hamiltonicity in $K_{r+1}$-free graphs. We
characterize the extremal graphs. Then we extend these edge density conditions
to clique density conditions. Equivalently, we introduce variants of the
extremal number $\mathrm{ex}(n,F)$ and the generalized extremal number
$\mathrm{ex}(n,H,F)$ -- that is, the maximum numbers of edges and copies of
$H$, respectively, in $n$-vertex, $F$-free graphs -- in which we require that
these graphs not be traceable, Hamiltonian, chorded pancyclic,
Hamiltonian-connected, $k$-path Hamiltonian, or $k$-Hamiltonian, and we
determine their values for $F=K_{r+1}$ and $H=K_t$.
