SOLVED
Let $G$ be a graph with chromatic number $\chi(G)=4$. If $m_1<m_2<\cdots$ are the lengths of the cycles in $G$ then can $\min(m_{i+1}-m_i)$ be arbitrarily large? Can this happen if the girth of $G$ is large?
The answer is no: Bondy and Vince
[BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$, and hence the same is true for every graph with chromatic number $4$.
