Flexible list colorings: Maximizing the number of requests satisfied

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose 0 ≤ ϵ ≤ 1, G is a graph, L is a list assignment for G, and r is a function with nonempty domain D ⊆ V ( G ) such that r ( v ) ∈ L ( v ) for each v ∈ 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 ∈ 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.