连通次立方图和立方图中支配-填充比的新下界
New lower bounds on domination--packing ratios in connected subcubic and cubic graphs
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中文总结 AI 辅助
该研究针对连通次立方图和立方图，构造显式二元分支族，证明其支配数与填充数之比的下界分别为13/6和17/8，反驳了\textit{γ(G)≤2ρ(G)+1}对连通立方图的适用性。
中文摘要 AI 辅助
对于图\textit{G}，令\textit{γ(G)}和\textit{ρ(G)}分别表示其支配数和填充数；令\textit{c_sub}和\textit{c_cub}分别表示当\textit{ρ(G)→∞}时，连通次立方图和连通立方图的\textit{γ(G)/ρ(G)}的上极限。我们通过构造两个显式的二元分支族证明：\textit{c_sub≥13/6}，\textit{c_cub≥17/8}。连通非立方次立方图\textit{ŷB_t^⋆}满足\textit{|V(ŷB_t^⋆)|=76·2^t−12}，\textit{γ(ŷB_t^⋆)=26·2^t−4}，\textit{ρ(ŷB_t^⋆)=12·2^t−2}；连通立方图\textit{ŷB_t^•}满足\textit{|V(ŷB_t^•)|=108·2^t−14}，\textit{γ(ŷB_t^•)=34·2^t−4}，\textit{ρ(ŷB_t^•)=16·2^t−2}。这些构造使用相同的二元连接器组合与闭合引理，仅连接器和初始集合不同。由此，两个族均给出\textit{γ(G)≤2ρ(G)+1}的无界加性违反，甚至反驳了该不等式对连通立方图的适用性。
英文摘要
For a graph \(G\), let \(γ(G)\) and \(ρ(G)\) denote its domination number and packing number, respectively. Let \(c_{\mathrm{sub}}\) and \(c_{\mathrm{cub}}\) denote the respective limsups of \(γ(G)/ρ(G)\) over connected subcubic and connected cubic graphs as \(ρ(G)\to\infty\). We prove \[ c_{\mathrm{sub}}\geq\frac{13}{6}, \qquad c_{\mathrm{cub}}\geq\frac{17}{8}, \] by constructing two explicit binary branching families. The connected noncubic subcubic graphs \(\widehat B_t^\star\) satisfy \[ |V(\widehat B_t^\star)|=76\cdot2^t-12,\qquad γ(\widehat B_t^\star)=26\cdot2^t-4,\qquad ρ(\widehat B_t^\star)=12\cdot2^t-2, \] whereas the connected cubic graphs \(\widehat B_t^\bullet\) satisfy \[ |V(\widehat B_t^\bullet)|=108\cdot2^t-14,\qquad γ(\widehat B_t^\bullet)=34\cdot2^t-4,\qquad ρ(\widehat B_t^\bullet)=16\cdot2^t-2. \] 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 \(γ(G)\leq2ρ(G)+1\), disproving the proposed inequality even for connected cubic graphs.
