Triangle-free 5-regular graphs of chromatic number five

Determine whether there exists a triangle-free 5-regular graph with chromatic number 5.

Background

The paper describes the problem of maximizing the chromatic number of a triangle-free graph with maximum degree Δ. It notes that the case Δ=5 remains unresolved: Grünbaum conjectured the existence of a triangle-free 5-regular graph requiring five colours, while other researchers conjectured the opposite. The question therefore concerns the exact extremal chromatic behaviour at maximum degree five.

References

Even the case $\Delta=5$ remains open: in 1970, Gr\"unbaum conjectured that there exists some triangle-free $5$-regular graph of chromatic number $5$. While others conjectured the opposite (see), it remains open to debate.

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Section “Chromatic number of triangle-free graphs”