k-forested choosability of graphs with bounded maximum average degree

TL;DR

This paper investigates the $k$-forested list coloring properties of graphs with bounded maximum average degree, establishing upper bounds on their $k$-forested choosability based on maximum degree and average degree constraints.

Contribution

It provides new upper bounds on the $k$-forested choosability of graphs with bounded maximum average degree, extending understanding of graph coloring under these conditions.

Findings

Upper bounds depend on maximum degree and average degree thresholds.

For graphs with max average degree less than 12/5, choosability is at most $oxed{igl\u2308rac{ ext{max degree}}{k-1}igr ceil+1}$.

Similar bounds are established for higher average degree thresholds.

Abstract

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prove that the $k$-forested choosability of a graph with maximum degree $\Delta\geq k\geq 4$ is at most $\lceil\frac{\Delta}{k-1}\rceil+1$, $\lceil\frac{\Delta}{k-1}\rceil+2$ or $\lceil\frac{\Delta}{k-1}\rceil+3$ if its maximum average degree is less than 12/5, $8/3 or 3, respectively.