AI 中文总结
研究可匹配的平面3-连通图中完美匹配数量，通过证明至少40个顶点的此类图至少有12个完美匹配且下界紧，否定了是否存在常数\(c<12\)使有无穷多此类图恰有\(c\)个完美匹配的问题。
AI 中文摘要
若图存在完美匹配，则称其为可匹配的。最近，Goedgebeur等人询问是否存在常数\(c<12\)，使得有无穷多个可匹配的平面3-连通图，每个图恰好有\(c\)个完美匹配。我们证明，每个至少有40个顶点的可匹配平面3-连通图至少有12个完美匹配，且此下界是紧的。这否定地回答了该问题。
英文摘要
A graph is matchable if it admits a perfect matching. Recently, Goedgebeur et al. asked whether there exists a constant $c<12$ such that infinitely many matchable planar $3$-connected graphs, each with exactly $c$ perfect matchings. We answer this question by proving that every matchable planar $3$-connected graph on at least 40 vertices has at least 12 perfect matchings, and this lower bound is sharp.
Comments12 pages,5 figures
