Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers

Source abstract

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 $ω=χ$.