最大度至多为7的图的Tuza猜想
Tuza's conjecture for graphs of maximum degree at most seven
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中文总结 AI 辅助
该研究证明最大度至多为7的图满足Tuza猜想，通过改进Puleo的可归约集框架，结合机器验证的局部证书完成极小7-正则反例的分析，明确了猜想常数的最优性。
中文摘要 AI 辅助
Tuza猜想指出，每个有限简单图G都满足τ(G)≤2ν(G)，其中ν(G)是两两边不交三角形的最大数目，τ(G)是删除后使G无三角形的最少边数。Puleo证明了最大平均度小于7的图均满足该猜想，这覆盖了最大度至多为6的图，但不包含7-正则图。我们证明了最大度至多为7的图满足该猜想。证明采用了Puleo的可归约集框架：在平均度为7时，其放电步骤不再强制产生可归约构型；在一个极小7-正则反例中，每个顶点的关联图是弱Konig-Egervary类之外的连通7顶点图，对这类关联图的穷尽枚举表明，每个顶点处存在一条关联边，位于4、5或6个三角形中。我们证明其端点构成可归约对：余度为5和6时使用打包与覆盖模板及Fano平面见证，余度为4时使用含1144个经机器验证的局部证书的显式目录，我们未提供该目录的人类可读证明，证书及其验证器随论文一同提供。常数2在最大度为3时已达到最优。
英文摘要
Tuza conjectured that every finite simple graph $G$ satisfies $τ(G) \leq 2ν(G)$, where $ν(G)$ is the maximum number of pairwise edge-disjoint triangles and $τ(G)$ is the minimum number of edges whose deletion makes $G$ triangle-free. Puleo proved the conjecture for every graph of maximum average degree less than $7$; this covers maximum degree at most $6$ but no $7$-regular graph. We prove the conjecture for maximum degree at most $7$. The proof uses Puleo's reducible-set framework. At average degree seven his discharging step no longer forces a reducible configuration. In a minimal $7$-regular counterexample every vertex link is a connected seven-vertex graph outside the weak Konig-Egervary class. An exhaustive census of such links supplies, at every vertex, an incident edge lying in four, five or six triangles. We prove that its endpoints form a reducible pair: codegrees five and six use a packing and covering template and Fano-plane witnesses, while codegree four uses an explicit catalogue of 1,144 machine-checked local certificates. We do not provide a human-readable proof of that catalogue; the certificates and their verifiers accompany the paper. The constant $2$ is sharp already at maximum degree three.
