Find an example of a regular triangle-free $4$-chromatic graph Find an example of a regular triangle-free $4$-chromatic graph
I know that for every $k \geq 3$ there exists a triangle-free $k$-chromatic graph.
So if I can find a triangle-free graph $H$ such that $\chi(H)=3$, then I can use the Mycielski construction to obtain a triangle-free graph $G$ such that $\chi(G)=4$. However, the regular part keep getting me stuck. I try some odd cycle, I also tried the Petersen graph but still can't get a regular triangle-free $4$-chromatic graph.
I wonder if anyone can give me a hint, please.

Another example is given by Kneser graphs $K(n,k)$ with suitable parameters.
By Lovasz' theorem, the chromatic number of $K(n,k)$ is given by $n-2k+2$.
Moreover, if $n<3k$ we have that $K(n,k)$ is triangle-free, so:

$\color{red}{K(8,3)}$ is a triangle-free, $10$-regular graph on $56$ vertices with chromatic number $\chi=4$.

However, the minimal example of a triangle-free, regular graph with $\chi=4$ is given by another "topological graph", the Clebsch graph, with $16$ vertices and degree $5$.
