Some orientation theorems for restricted DP-colorings of graphs

TL;DR

This paper introduces new orientation theorems for restricted DP-colorings of graphs, generalizing existing concepts and providing tools for analyzing colorings on complex graph structures like toroidal grids.

Contribution

It defines Z-signable correspondence assignments, develops an Alon-Tarsi style theorem for DP-colorings, and applies these results to specific graph classes.

Findings

Established an Alon-Tarsi style theorem for Z-signable correspondence assignments.

Derived three DP-coloring analogs of the Alon-Tarsi theorem.

Applied the theorems to DP-colorings on toroidal grid graphs.

Abstract

We define $Z$-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from $Z$-signable correspondence assignments on multigraphs, and apply this theorem to obtain three DP-coloring analogs of the Alon-Tarsi theorem for arbitrary correspondence assignments as corollaries. We illustrate the use of these corollaries for DP-colorings on a restricted class of correspondence assignments on toroidal grids.