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

TL;DR

This paper investigates the DP-chromatic number of complete bipartite graphs, revealing new bounds that differ from the list chromatic number, and provides specific thresholds for these parameters.

Contribution

It establishes new bounds for the DP-chromatic number of complete bipartite graphs, showing how it diverges from the list chromatic number with explicit thresholds.

Findings

_{DP}(K_{k,t}) = k+1 for t ; _{DP}(K_{k,t}) < k+1 for smaller t

The known threshold for list chromatic number is t ; the DP-chromatic number threshold is different and explicitly calculated

The paper demonstrates that DP-chromatic number can be strictly less than the list chromatic number for certain bipartite graphs.

Abstract

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 $t < k^k/k!$.