Fractional DP-coloring of K_{t,t,t}-free graphs

Determine whether every K_{t,t,t}-free graph G with maximum degree Delta satisfies chi_DP(G) = O_t(Delta/log Delta).

Background

The paper derives a fractional DP-chromatic upper bound of order t2 Delta/log Delta for K_{t,t,t}-free graphs by exploiting a locally sparse vertex ordering. The authors note that, for ordinary DP-coloring, no nontrivial bound beyond the trivial Delta+1 estimate was known in the stated context.

The unresolved issue is whether the usual Delta/log Delta improvement holds for DP-coloring of all K_{t,t,t}-free graphs, with a constant depending only on t.

References

Determining whether $\chi{\mathrm{DP}(G) = O_t\left(\frac{\Delta}{\log \Delta}\right)$ for all $K_{t, t, t}$-free graphs $G$ remains an open problem.

— Fractional DP-colorings of $d$-degenerate locally sparse graphs  (2609.10978 - Dhawan et al., 10 Sep 2026) in Section 5, Concluding Remarks