This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse $d$-regular \emph{random} graphs, in the above proof systems, with high probability requires proofs with degree $Ω(n/\log n)$. We extend their result by showing the same lower bound holds for \emph{all} $d$-regular graphs with a mild spectral gap.

翻译：本研究探讨了在多项式演算（PC）与平方和（SoS）证明系统中，反驳具有奇数个顶点的谱扩展图中存在完美匹配的复杂性。Austrin与Risse [SODA, 2021] 指出，在上述证明系统中，以高概率反驳稀疏$d$-正则\emph{随机}图中存在完美匹配，需要证明具有$Ω(n/\log n)$的度。我们通过证明相同的下界对于所有具有温和谱间隙的$d$-正则图均成立，从而扩展了他们的结果。