On list

Let L be list assignment of colors available for vertices of a graph G, an ( L , r )-coloring of G is a proper coloring c such that for any vertex v ∈ V ( G ), we have c ( v ) ∈ L ( v ) and | c ( N ( v ) ) | ≥ min { d ( v ) , r }. The r- hued list chromatic number of G, denoted as χ L , r ( G ), is the least integer k, such that for list assignment L satisfying | L ( v ) | = k ∀ v ∈ V ( G ), G has an ( L , r )-coloring. A graph G is an outer-1-planar graph if G has a drawing on the plane such that all vertices of V ( G ) are located on the outer face of this drawing, and each edge can cross at most one other edge. For any positive integer r, we completely determine the upper bound of list r-hued chromatic number for all outer-1-planar graphs. This extended a former result in [Discrete Mathematics and Theoretical Computer Science, 23:3 (2021)].