Completing Partial Proper Colorings using Hall's Condition

In the context of list-coloring the vertices of a graph, Hall’s condition is a generalization of Hall’s Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph G with list assignment L satisfies Hall’s condition if for each subgraph H of G, the inequality |V (H)| 6∑σ∈C α(H(σ, L)) is satisfied, where C is the set of colors and α(H(σ, L)) is the independence number of the subgraph of H induced on the set of vertices having color σ in their lists. A list assignment L to a graph G is called Hall if (G,L) satisfies Hall’s condition. A graph G is Hall m-completable for some m > χ(G) if every partial proper m-coloring of G whose corresponding list assignment is Hall can be extended to a proper coloring of G. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall m-completable, but not Hall (m+ 1)-completable for some m > 3? (2) If G is neither complete nor an odd cycle, is G Hall ∆(G)-completable? This paper establishes that for every m > 3, there exists a graph that is Hall mcompletable but not Hall (m+ 1)-completable and also that every bipartite planar graph G is Hall ∆(G)-completable.