We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape $C$. Let $S$ be a point set in the plane. The $k$-order Delaunay graph of $S$, denoted $k-DGC(S)$, has vertex set $S$ and edge $pq$ provided that there exists some homothet of $C$ with $p$ and $q$ on its boundary and containing at most $k$ points of $S$ different from $p$ and $q$. The $k$-order Gabriel graph $k-GGC(S)$ is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of $C$ with $p$ and $q$ on its boundary.
We provide upper bounds on the minimum value of $k$ for which $k-GGC(S)$ is Hamiltonian. Since $k-GGC(S)\subseteq k-DGC(S)$, all results carry over to $k-DGC(S)$. In particular, we give upper bounds of 24 for every $C$ and 15 for every point-symmetric $C$. We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular $t$-gons (for $t\geq 10$). These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs.