A Note on the DP-Chromatic Number of Complete Bipartite Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo\v{r}\'{a}k and Postle. Several known bounds for the list chromatic number of a graph $G$, $\chi_\ell(G)$, also hold for the DP-chromatic number of $G$, $\chi_{DP}(G)$. On the other hand, there are several properties of the DP-chromatic number that shows that it differs with the list chromatic number. In this note we show one such property. It is well known that $\chi_\ell (K_{k,t}) = k+1$ if and only if $t \geq k^k$. We show that $\chi_{DP} (K_{k,t}) = k+1$ if $t \geq 1 + (k^k/k!)(\log(k!)+1)$, and we show that $\chi_{DP} (K_{k,t}) < k+1$ if $t < k^k/k!$.