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.

The Gewirtz graph, a strongly regular graph with parameters $(56,10,0,2)$, is another triangle-free graph with chromatic number $\chi=4$.
It is worth noting that Gewirtz graph is not isomorphic to Kneser graph $K(8,3)$. The Gewirtz graph has independence number $\alpha(G) = 16$ (i.e. the cardinality of a largest set of vertices pairwise nonadjacent, e.g. see Brouwer and Van Maldeghem' survey) whereas $\alpha(K(8,3)) = 21$ (the Erdős-Ko-Rado theorem  states that $\alpha(K(n,k)) = \binom{n-1}{k-1}, n\geq 2k $).
