# Choosability in bounded sequential list coloring

## TL;DR

This paper explores the complexity of list coloring in graphs with sequential color constraints, showing that certain choosability problems are NP-complete and analyzing specific graph classes.

## Contribution

It establishes the NP-completeness of $k$-$(eta,
u)$-choosability and analyzes choosability properties for specific graph classes like complete bipartite graphs.

## Key findings

- $k$-$(eta,
u)$-choosability is NP-complete.
- Complete bipartite graphs are 3-choosable but 2-$(eta,
u)$-choosable.
- Choosability complexity varies with graph class and list constraints.

## Abstract

The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring, where the color lists have sequential values with known lower and upper bounds. This work discusses the choosability property, that consists in determining the least number $k$ for which it has a proper list coloring no matter how one assigns a list of $k$ colors to each vertex. This is a $\Pi_2^P$-complete problem, however, we show that $k$-$(\gamma,\mu)$-choosability is an $NP$-problem due to its relation with the $k$-coloring of a graph and application of methods of proof in choosability for some classes of graphs, such as complete bipartite graph, which is $ 3 $-choosable, but $ 2 $-$(\gamma,\mu)$-choosable.