Every Graph G is Hall Δ(G)-Extendible

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, abbreviated (G,L), satisfies Hall’s condition if for each subgraph H of G, the inequality |V (H)| 6σ∈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 k-extendible for some k > χ(G) if every k-precoloring of G whose corresponding list assignment is Hall can be extended to a proper k-coloring of G. In 2011, Bobga et al. posed the question: If G is neither complete nor an odd cycle, is G Hall ∆(G)-extendible? This paper establishes an affirmative answer to this question: every graph G is Hall ∆(G)-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of k for which they are Hall k-extendible is presented. We include a focus on graphs which are Hall k-extendible for all k > χ(G), since these are graphs for which satisfying the obviously necessary Hall’s condition is also sufficient for a precoloring to be extendible.