发表机构
University of Pannonia; HUN-REN Rényi Institute of Mathematics（潘诺尼亚大学; 匈牙利研究网络雷尼数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出几乎堆叠猜想（ASH），证明其对偶圈和树的堆叠与清除数成立，并构造边密度趋于零的完美可卵石图无限族，表明高边密度并非必要条件。
AI 中文摘要
我们通过几乎堆叠猜想（ASH）研究两个图卵石参数：堆叠数和清除数。该猜想断言，这些阈值可以通过仅测试至多一个顶点携带多于一个卵石的配置来确定。我们证明了在$C_{2n}$上大小为$2^{n+1}-1$的每个几乎堆叠配置都是可堆叠的，并且在$C_{2n+1}$上大小为$3\cdot 2^n-2$的每个几乎堆叠配置都是可清除的。结合已知的下界，这些结果表明ASH蕴含$\operatorname{stack}(C_{2n})=2^{n+1}-1$和$\operatorname{clear}(C_{2n+1})=3\cdot 2^n-2$。对于有限树$T$，我们引入一个显式不变量$\operatorname{estim}(T)$。我们无条件证明了$\operatorname{stack}(T)\geq\operatorname{estim}(T)$，并在ASH下证明了反向不等式。因此，ASH得出$\operatorname{stack}(T)=\operatorname{estim}(T)$，我们猜想该等式无条件成立。最后，我们研究完美可卵石图：清除数具有最小可能值$\operatorname{clear}(G)=|V(G)|+1$的有限连通非二分图。每个至少有三个顶点的完全图都是完美可卵石的，这可能表明完美可卵石性需要高边密度。然而，假设ASH，我们表明情况并非如此。我们给出了一个涉及强边三角剖分以及顶点删除子图中的哈密顿路径和路径覆盖条件的充分准则，并利用它构造了两个边密度趋于零的完美可卵石图的显式无限族，其中一个只有线性数量的边。
英文摘要
We study two graph pebbling parameters, the stacking number and the clearing number, through the Almost Stacked Hypothesis (ASH). This hypothesis asserts that these thresholds can be determined by testing only configurations in which at most one vertex carries more than one pebble. We prove that every almost stacked configuration of size $2^{n+1}-1$ on $C_{2n}$ is stackable and that every almost stacked configuration of size $3\cdot 2^n-2$ on $C_{2n+1}$ is clearable. Together with the known lower bounds, these results show that ASH implies $\operatorname{stack}(C_{2n})=2^{n+1}-1$ and $\operatorname{clear}(C_{2n+1})=3\cdot 2^n-2$. For a finite tree $T$, we introduce an explicit invariant $\operatorname{estim}(T)$. We prove unconditionally that $\operatorname{stack}(T)\geq\operatorname{estim}(T)$ and prove the reverse inequality under ASH. Consequently, ASH yields $\operatorname{stack}(T)=\operatorname{estim}(T)$, and we conjecture that this equality holds unconditionally. Finally, we study perfectly pebblable graphs: finite connected non-bipartite graphs whose clearing number has the minimum possible value $\operatorname{clear}(G)=|V(G)|+1$. Every complete graph with at least three vertices is perfectly pebblable, which might suggest that perfect pebblability requires high edge density. Assuming ASH, however, we show that this is not the case. We give a sufficient criterion involving strong edge-triangulation and Hamiltonian-path and path-cover conditions in vertex-deleted subgraphs and use it to construct two explicit infinite families of perfectly pebblable graphs with edge density tending to zero, one of which has only a linear number of edges.
Comments23 pages
