A Note on Partial List Colorings

TL;DR

This paper investigates partial list coloring in graphs, proving a conjecture for at least half of the list sizes and proposing a new related conjecture with supporting results.

Contribution

The paper confirms a conjecture for many list sizes and introduces a new related conjecture in the field of partial list coloring.

Findings

Conjecture holds for at least half of the list sizes

New partial list coloring results support the conjecture

Proposes and explores a new related conjecture

Abstract

Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that at least $\frac{tn}{\chi_\ell}$ vertices of $G$ can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least half of the numbers of the set $\{1,2,...,\chi_\ell(G)-1\}$. In addition we introduce a new related conjecture and finally we present some results about this conjecture.