AI 中文总结
该研究突破BBF预测，构造78顶点非同构图对有51张公共卡片，还对任意偶数r≥4构造图族使公共卡片比例渐近≥1-1/r，证明非同构图可共享任意大比例顶点删除卡片。
AI 中文摘要
对于图G，其顶点 deck 是删除一个顶点后得到的图的多重集。Bowler、Brown 和 Fenner（BBF）提出，对于所有足够大的n，两个非同构n顶点图的 deck 之间的最大可能重叠为2⌊(n-1)/3⌋。我们首先给出一对78顶点的连通非同构图，其至少有51张公共卡片，超过了BBF预测的50。接着，我们对每个偶数r≥4，构造出任意大阶数的图族，其重叠比例渐近至少为1-1/r。因此，对于每个α<1，存在无穷多对图有超过αn张公共卡片，故可达到的比例任意接近完整 deck。对于代表性实例，我们还使用Brendan McKay的nauty工具，通过完整deck生成和同构测试验证了预测的重叠值。
英文摘要
For a graph $G$, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed $2\lfloor(n-1)/3\rfloor$ as the maximum possible overlap between the decks of two nonisomorphic $n$-vertex graphs, for all sufficiently large $n$. We first give an explicit pair of connected nonisomorphic graphs on $78$ vertices with at least $51$ common cards, exceeding BBF's predicted value of $50$. We then construct, for every even $r\ge4$, families at arbitrarily large orders whose overlap fraction is asymptotically at least $1-1/r$. Consequently, for every $α<1$, infinitely many pairs have more than $αn$ common cards, so the attainable fraction is arbitrarily close to the full deck. For representative instances, the predicted overlaps were also checked by complete deck generation and isomorphism testing with Brendan McKay's nauty tools.
