Cayley 竞赛图同时关于团数和二色数临界
Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers
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中文总结 AI 辅助
本文提出模板提升构造，生成同时关于团数和二色数临界的正则顶点传递 Cayley 竞赛图，证明相关猜想并否定有界证书问题，同时刻画循环替换的团数与临界性。
中文摘要 AI 辅助
对于竞赛图 $T$，设 $\omega(T)$ 为 $T$ 的返边图（backedge graph）中最小团数，设 $\chi(T)$ 为其二色数。我们给出一个模板提升（template-lifting）构造。该构造将一个 $k$-模板转化为一个正则的、顶点传递的 Cayley 竞赛图，该图同时是 $(k+1)$-$\omega$-临界和 $(k+1)$-$\chi$-临界的。输出也是一个 $(k+1)$-模板。迭代该构造，我们证明对于每个 $k\geq3$，存在一个正偶数 $m_k$ 满足以下性质：每个 $N>1$ 且 $N\equiv1\pmod{m_k}$ 的整数 $N$ 都是一个正则的、顶点传递的 Cayley 竞赛图的阶，该图同时是 $k$-$\omega$-临界和 $k$-$\chi$-临界的。这证明了 Aboulker、Aubian、Charbit 和 Lopes 的一个猜想，并在假设 $\omega(T)\geq k$ 时，对他们的有界证书问题给出了否定答案。我们还找到了当每个块满足 $\omega=\chi$ 时循环替换（cyclic substitution）的团数。然后，当块是 $\chi$-临界且满足 $\omega=\chi$ 时，我们精确描述了该替换何时是 $\omega$-临界的。
英文摘要
For a tournament $T$, let $ω(T)$ be the minimum clique number among the backedge graphs of $T$, and let $χ(T)$ be its dichromatic number. We give a template-lifting construction. It turns a $k$-template into a regular, vertex-transitive Cayley tournament that is simultaneously $(k+1)$-$ω$-critical and $(k+1)$-$χ$-critical. The output is also a $(k+1)$-template. Iterating the construction, we prove that for every $k\geq3$, there is a positive even integer $m_k$ with the following property. Every $N>1$ with $N\equiv1\pmod{m_k}$ is the order of a regular, vertex-transitive Cayley tournament that is simultaneously $k$-$ω$-critical and $k$-$χ$-critical. This proves a conjecture of Aboulker, Aubian, Charbit, and Lopes and gives a negative answer to their bounded-certificate question when the hypothesis is $ω(T)\geq k$. We also find the clique number of a cyclic substitution when each block satisfies $ω=χ$. We then describe exactly when this substitution is $ω$-critical if the blocks are $χ$-critical and satisfy $ω=χ$.
