一个50顶点的三次图反例：支配数对边支配数猜想
A 50-Vertex Cubic Counterexample to the Domination-versus-Edge-Domination Conjecture
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中文总结 AI 辅助
本文通过一个50顶点三次图反例，证明Baste等人关于正则图支配数不超过边支配数的猜想不成立，并给出精确计算与验证。
中文摘要 AI 辅助
Baste、Furst、Henning、Mohr和Rautenbach猜想：每个有限正则图（正度数）满足\\(\gamma(G) \leq \gamma_e(G)\\)，其中\\(\gamma\\)是支配数，\\(\gamma_e\\)是边支配数，等价于最大匹配的最小基数。我们证明该猜想对三次图（即每个顶点度数为3的图）已经不成立。反例是一个先前公开的50顶点三次图，该图曾被用于反驳更强的独立支配不等式\\(i(G) \leq \gamma_e(G)\\)。对于该图，我们证明\\(\gamma(G) = 16 > 15 = \gamma_e(G)\\)。等式\\(\gamma_e(G) = 15\\)有一个简短的计数证明，并且显式给出了一个大小为16的支配集。对于下界\\(\gamma(G) \geq 16\\)，我们给出一个自包含的精确归约：在固定20个子句顶点中哪些属于一个假定的支配集之后，剩余问题是在30个文字顶点上的一个有限集合覆盖问题。我们枚举所有\\(2^{20} = 1,048,576\\)个子句子集，推导出两个显式下界，并通过论文中陈述的递推精确求解5,931个剩余情形。展示了完整的案例计数和最小值，并在附录中包含一个简短的Python标准库实现。一个独立的893,049节点证明树证书和直接图搜索提供了独立验证。因此，正则图猜想被推翻。结合Gupta最近关于每个至多48个顶点的三次图满足猜想不等式的定理，该例子是三次反例中阶最小的。
英文摘要
Baste, Furst, Henning, Mohr, and Rautenbach conjectured that every finite regular graph of positive degree satisfies \(γ(G) \leq γ_e(G)\), where \(γ\) is the domination number and \(γ_e\) is the edge domination number, equivalently the minimum cardinality of a maximal matching. We show that the conjecture is false already for cubic graphs. The counterexample is a previously public 50-vertex cubic graph that had been used to refute the stronger independent-domination inequality \(i(G) \leq γ_e(G)\). For this graph we prove \(γ(G) = 16 > 15 = γ_e(G)\). The equality \(γ_e(G) = 15\) has a short counting proof, and a dominating set of order 16 is displayed explicitly. For the lower bound \(γ(G) \geq 16\), we give a self-contained exact reduction: after fixing which of the 20 clause vertices lie in a putative dominating set, the remaining problem is a finite set-cover problem on the 30 literal vertices. We enumerate all \(2^{20} = 1,048,576\) clause subsets, derive two explicit lower bounds, and solve exactly the 5,931 residual cases by a recurrence stated in the paper. The complete case counts and minima are displayed, and a short standard-library Python implementation is included in an appendix. A separate 893,049-node proof-tree certificate and a direct graph search provide independent verification. Thus the regular-graph conjecture is disproved. Combined with Gupta's recent theorem that every cubic graph on at most 48 vertices satisfies the conjectured inequality, the example is order-minimal among cubic counterexamples.
