The $2t+2$-cycle conjecture for chromatic numbers of graph powers

Let $G$ be a graph of maximum degree at most $d$, and let $G^t$ denote its $t$th power. Suppose that $G$ contains no cycle of length $2t+2$ as a subgraph. The $2t+2$-cycle conjecture. The largest possible value of the chromatic number $\chi(G^t)$, over all such graphs $G$, is

$$\Theta(d^t/\log d)$$

as $d\to\infty$. The conjecture concerns the threshold immediately below the critical-girth phenomenon of Alon and Mohar and would extend the paper's logarithmic upper-bound results; it remains open in the stated generality.