发表机构
New Uzbekistan University（新乌兹别克斯坦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了两类全新无限族完全图的列表边着色猜想，通过互补模方法控制1-因子分解的Pfaffian符号，建立行列式-Pfaffian桥梁并发展带符号Burnside-Fourier方法，同时揭示了该方法在高层论证中的赋值障碍。
AI 中文摘要
设p为奇素数，我们证明了两类无限族完全图的列表边着色猜想：χ'_ℓ(K_{p-1})=p-2，χ'_ℓ(K_{2p})=2p-1。两个证明通过互补模方法控制1-因子分解的Pfaffian符号：对于K_{p-1}，Frobenius映射和斜特殊化将Glynn的行列式系数同余转化为无平方因子的Pfaffian系数，再通过差商将剩余计算简化为单个反对角线Pfaffian，得到[x^1]Pf(X)^{p-2}≡(-2)^{(p-1)/2} mod p；对于K_{2p}，对平移群F_p²的加权Burnside计数分离出带符号循环起始项和，利用斜循环子式恒等式计算其平方，得到S_{2p}≡-p mod p²。特别地，两个决定性带符号和均非零，且这两个同余均非拉丁方奇偶性的形式结果——二分图行列式符号与非二分图Pfaffian符号是不同不变量，相反，证明建立了行列式-Pfaffian桥梁，并发展了适配完全图符号的带符号Burnside-Fourier方法。我们还找到了该方法的一个极限：对所有偶数b≥4，K_{bp}上全支撑平移的带符号迹可被p^b整除，当b=4时这意味着p^4|S_{4p}但未给出非零剩余，揭示了全支撑高层论证的赋值障碍。
英文摘要
The List Edge-Coloring Conjecture predicts that any graph whose edges can be colored with $k$ colors can also be colored from arbitrary lists of $k$ colors. We prove its stronger online form for two new infinite families, $K_{p-1}$ and $K_{2p}$, where $p$ is an odd prime. For even $n$, order the vertices of $K_n$ and draw each perfect matching as arcs above them. Count crossings separately within each matching, and let $S_n$ be the number of decompositions into perfect matchings having an even total crossing count minus the number having an odd total. Then \[ S_{p-1}\equiv\left(\frac{-2}{p}\right)\pmod p, \qquad S_{2p}\equiv-p\pmod {p^2}. \] The two congruences are governed by the same elementary matching sum over $\F_p$, although their proofs use the prime $p$ differently. Their nonzero residues give the conjectured values even in the online game. They also treat the corresponding complete graphs with one perfect matching removed, as well as $K_{2p}$ after deleting some, but not all, of a natural cyclic family of $p$ disjoint perfect matchings.
