Flexible list colorings: Maximizing the number of requests satisfied

TL;DR

Improve a result of Dvo\v{r}\'{a}k et al., by showing $d-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible, and study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

Abstract

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose $0 \leq ε\leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). The triple $(G,L,r)$ is $ε$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $ε|D|$ vertices in $D$. We say $G$ is $(k, ε)$-flexible if $(G,L',r')$ is $ε$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It was shown by Dvořák et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs. The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $ε> 0$ such that $G$ is $(k, ε)$-flexible, but it is natural to try to find the largest possible $ε$ for which $G$ is $(k,ε)$-flexible. In this vein, we improve a result of Dvořák et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $ε$ for which a graph is $(k,ε)$-flexible, we observe that a graph $G$ is not $(k, ε)$-flexible for any $k$ if and only if $ε> 1/ ρ(G)$, where $ρ(G)$ is the Hall ratio of $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ ρ(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

Theorems & Definitions (39)

• Theorem 1: DN19
• Theorem 2: DN19
• Theorem 3: BM22
• Theorem 4
• Proposition 5
• Corollary 6
• Proposition 7
• Proposition 8
• proof
• Proposition 13