AI 中文总结
本文研究匹配-拼图实现问题的复杂性，针对稠密与稀疏图给出多项式算法、NP完全性阈值及计数复杂性结果。
AI 中文摘要
匹配-拼图问题询问：能否对给定图的顶点进行着色，使得其边所诱导的颜色对多重集恰好等于一个预先指定的多重集。我们研究该实现问题的复杂性如何依赖于宿主图。在稠密侧，对于任意固定部数$k$的完全$k$部图，我们给出了一个多项式时间算法，允许任意预着色和任意数量的颜色。我们证明了一个尖锐的补图度阈值：当$\Delta(\overline G)\le1$时，问题可在多项式时间内求解，但当$\Delta(\overline G)=2$时，问题在完全未着色图上已是NP完全的。这产生了均匀完全多部图的一个二分性，且连通直径二已足以导致NP完全性。我们还证明了在余图上以颜色数为参数的W[1]困难性。在稀疏侧，完全未着色的路径和圈允许通过欧拉迹和欧拉回路进行线性时间刻画，而在这两类图上计数可行着色是$\\\\#P$-完全的。尽管如此，在星图和完全图上，即使有任意预着色，计数仍可在多项式时间内求解。一个分解-传递定理在树深度二时即产生NP完全性。一个独立的路径分解归约给出了完全未着色且颜色数不受限制的不连通宿主图的最大度阈值在一和二之间。边数至多为二的连通分量是可处理的，而$P_4$的不交并是NP完全的。最后，预着色在若干情况下恢复了可处理性：当每个中心都被预着色时，星森林是多项式的；在长度二蜘蛛上的两个广泛预着色体制下，即使颜色数无界，问题也是多项式的。
英文摘要
The Matching-Match puzzle asks whether the vertices of a fixed graph can be colored so that the multiset of color pairs induced by its edges is exactly a prescribed multiset. We study how the complexity of this realization problem depends on the host graph. On the dense side, we give a polynomial-time algorithm for complete $k$-partite graphs for every fixed number $k$ of parts, with arbitrary precoloring and an arbitrary number of colors. We prove a sharp complement-degree threshold: the problem is polynomial-time solvable when $Δ(\overline G)\le1$, but NP-complete on completely uncolored graphs already when $Δ(\overline G)=2$. This yields a dichotomy for uniform complete multipartite graphs, and connected diameter two already suffices for NP-completeness. We also prove W[1]-hardness on cographs parameterized by the number of colors. On the sparse side, completely uncolored paths and cycles admit a linear-time characterization by Euler trails and circuits, while counting feasible colorings is $\#P$-complete on both classes. Counting is nevertheless polynomial-time solvable on stars and complete graphs, even with arbitrary precoloring. A decomposition-transfer theorem yields NP-completeness already at tree-depth two. A separate path-decomposition reduction gives a maximum-degree threshold between one and two for completely uncolored disconnected host graphs with unrestrictedly many colors. Components with at most two edges are tractable, while a disjoint union of $P_4$'s is NP-complete. Finally, precoloring restores tractability in several cases: star forests are polynomial when every center is precolored, and two broad precoloring regimes on length-two spiders are polynomial even when the number of colors is unbounded.
