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.