AI 中文总结
研究Payan提出的有限(0,2)-图色数能否为三的问题，通过证明每个有限三色(0,2)-图是二分图，得出不存在色数恰好为三的有限(0,2)-图的结论。
AI 中文摘要
Payan的一个定理表明立方图的色数不可能恰好为三。一个相关问题，通常作为Payan的有限(0,2)-图问题来讨论，即询问每个两个不同顶点要么有零个要么有两个共同邻居的有限图的色数是否能恰好为三。有限性假设是有意义的，因为可以构造出无限的三色(0,2)-图。我们证明了每个有限三色(0,2)-图都是二分图。因此，不存在色数恰好为三的有限(0,2)-图。
英文摘要
A theorem of Payan says that a cubelike graph cannot have chromatic number exactly three. A nearby question, usually discussed as Payan's finite $(0,2)$-graph question, asks whether a finite graph in which every two distinct vertices have either zero or two common neighbours can have chromatic number exactly three. The finite hypothesis is meaningful: infinite three-chromatic $(0,2)$-graphs can be constructed \cite{Payan1992}. We prove that every finite three-colourable $(0,2)$-graph is bipartite. Thus, no finite $(0,2)$-graph has chromatic number exactly three.
