Flexible list colorings: Maximizing the number of requests satisfied

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose $0\le ϵ\le 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with nonempty domain $D\subseteq V\left(G\right)$ such that $r\left(v\right)\in L\left(v\right)$ for each $v\in D$ ($r$ is called a request of $L$). The triple $\left(G,L,r\right)$ is $ϵ$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f\left(v\right)=r\left(v\right)$ for at least $ϵ\mid D\mid$ vertices in $D$. We say $G$ is $\left(k,ϵ\right)$-flexible if $\left(G,L^{\prime },r^{\prime }\right)$ is $ϵ$-satisfiable whenever $L^{\prime }$ is a $k$-assignment for $G$ and $r^{\prime }$ is a request of $L^{\prime }$. 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 $\left(G,L,r\right)$ is 1-satisfiable whenever $L$ is a $\left(d+1\right)$-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 $k\in N$ there exists an $ϵ>0$ such that $G$ is $\left(k,ϵ\right)$-flexible, but it is natural to try to find the largest possible $ϵ$ for which $G$ is $\left(k,ϵ\right)$-flexible. In this vein, we improve a result of Dvořák et al., by showing $d$-degenerate graphs are $\left(d+2,1∕2^{d+1}\right)$-flexible. In pursuit of the largest $ϵ$ for which a graph is $\left(k,ϵ\right)$-flexible, we observe that a graph $G$ is not $\left(k,ϵ\right)$-flexible for any $k$ if and only if $ϵ>1∕\rho \left(G\right)$, where $\rho \left(G\right)$ 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 $\left(k,1∕\rho \left(G\right)\right)$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.