New lower bounds on domination--packing ratios in connected subcubic and cubic graphs

For a graph $G$, let $\gamma (G)$ and $\rho (G)$ denote its domination number and packing number, respectively. Let $c_{\mathrm{s}\mathrm{u}\mathrm{b}}$ and $c_{\mathrm{c}\mathrm{u}\mathrm{b}}$ denote the respective limsups of $\gamma (G)\mathrm{/}\rho (G)$ over connected subcubic and connected cubic graphs as $\rho (G)\to \mathrm{\infty }$. We prove $$c_{\mathrm{s}\mathrm{u}\mathrm{b}}\ge \frac{13}{6},c_{\mathrm{c}\mathrm{u}\mathrm{b}}\ge \frac{17}{8},$$ by constructing two explicit binary branching families. The connected noncubic subcubic graphs ${\widehat{B}}_t^{\star }$ satisfy $$\mathrm{\mid }V({\widehat{B}}_t^{\star })\mathrm{\mid }=76\cdot 2^t-12,\gamma ({\widehat{B}}_t^{\star })=26\cdot 2^t-4,\rho ({\widehat{B}}_t^{\star })=12\cdot 2^t-2,$$ whereas the connected cubic graphs ${\widehat{B}}_t^{\bullet }$ satisfy $$\mathrm{\mid }V({\widehat{B}}_t^{\bullet })\mathrm{\mid }=108\cdot 2^t-14,\gamma ({\widehat{B}}_t^{\bullet })=34\cdot 2^t-4,\rho ({\widehat{B}}_t^{\bullet })=16\cdot 2^t-2.$$ The constructions use the same binary connector composition and closing lemma, with different connectors and initial assemblies. As a consequence, both families give unbounded additive violations of $\gamma (G)\le 2\rho (G)+1$, disproving the proposed inequality even for connected cubic graphs.