Every graph G is Hall ∆ ( 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.


Introduction
Throughout, G is a finite, simple graph with vertex set V (G) and edge set E(G).For U ⊆ V (G), we shall use G[U ] to denote the subgraph of G induced on U .Additionally α(G), δ(G), ∆(G), χ(G), shall denote the independence number, minimum degree, maximum degree, and chromatic number of G respectively.Let deg G (v) denote the degree of the vertex v in the graph G.For any U ⊆ V (G) and any subgraph H of G, let N H (U ) denote the set of vertices in H that are adjacent to at least one vertex in U .Let [m] denote the set {1, . . ., m}.We refer the reader to West [16] for any notation not defined here.Definition 1.A k-precoloring of G is a proper k-coloring of G[U ] where U ⊂ V (G).The coloring, say φ, can be extended (or is extendible) if there exists a proper kcoloring θ : V (G) → [k] where θ(v) = φ(v) for all v ∈ U .
The Precoloring Extension Problem (PrExt) is a natural generalization of the usual graph coloring problem and has been heavily studied.Of course, any precoloring with ∆(G) + 1 colors can be extended greedily, but a precoloring with ∆(G) colors that gives each vertex in the neighborhood of a vertex v with deg G (v) = ∆(G) a different color cannot be extended.Therefore PrExt is only interesting if conditions are placed on the precoloring.Most results (see for example [1], [2], [3], [4], [14]) place distance-based conditions on the precolored set U ; the precolored vertices need to be "far enough apart." Our approach to the PrExt problem in this article is entirely different.We guarantee the extension of precolorings through an obvious necessary condition that is based on Hall's condition for matching extensions.As is common, this condition views the PrExt problem as a list coloring problem.Vizing [15] introduced the notion of list coloring.It was further developed by Erdős, Rubin, and Taylor [9], and has been studied extensively since.If C is an infinite set of colors (the palette) and L is a set of finite subsets of C, then a list assignment to G is a function L : V (G) → L. The list L is a k-assignment to G if |L(v)| k for all v ∈ V (G).Given a list assignment L of G with color palette C, an L-coloring of G is a function φ : V (G) → C such that φ(v) ∈ L(v) for every vertex v.An L-coloring φ is proper if each color class induces an independent set.If G has a proper L-coloring, we say G is L-colorable.
In 1990, Hilton and Johnson [11] introduced the following concept (also see [5]), which was a generalization of Philip Hall's 1935 Marriage Theorem ( [10]) applied to list assignments of graphs.Suppose that φ is an L-coloring of G for some list assignment L with a color palette C and let H be any subgraph of G.For each σ ∈ C, consider φ −1 (σ) | H , the set of all vertices in H given color σ under φ, and let H(σ, L) be the subgraph of H induced on all vertices of H having σ in their lists.Then φ −1 (σ) | H is an independent set of vertices contained inside H(σ, L).Naturally, if G is L-colorable, then for every subgraph H, we must have This motivated the following definition: Definition 2 ( [11]).The graph G with list assignment L satisfies Hall's condition if for each subgraph H of G, the inequality is satisfied.For brevity we say (G, L) satisfies Hall's condition provided that ( * ) is satisfied for every subgraph of G.If H is a subgraph of G, then (H, L) will denote the natural restriction of L to V (H).If (G, L) does not satisfy (i.e., fails) Hall's condition, then there exists some subgraph H of G such that (H, L) does not satisfy the inequality ( * ).While satisfying Hall's condition is a necessary condition for G to have a proper Lcoloring, it is not sufficient; see Figure 1 for an example of a list assignment L to a 4-cycle in which all subgraphs satisfy ( * ), but there is no proper L-coloring.
Verifying that a pair (G, L) satisfy Hall's condition is difficult, as generally all subsets of G must be considered.There are several results that assist in this verification process.
The following is proved from a simple extension of the proof of Theorem 4: Theorem 5.If L is a list assignment to a graph G and for every subgraph H, the average list cardinality when restricted to V (H) is at least χ(G), then (G, L) satisfies Hall's condition.
Proof.Suppose that for each H G, but (G, L) fails Hall's condition.Then for some subgraph K of G, we have a contradiction.The last inequality follows from the assumption that (1) holds for every subgraph of G. Definition 6. [11] The Hall number of G is the smallest positive integer h(G) = k such that whenever L is a k-assignment to G and (G, L) satisfies Hall's condition, G is L-colorable.
In other words, h(G) is the smallest positive integer such that Hall's condition on k-assignments is both necessary and sufficient for the existence of a proper L-coloring of G.
The following result characterizes graphs with Hall number 1.

3.
G contains no induced cycle C n , n 4, nor an induced copy of K 4 − e (that is, K 4 with an edge deleted).
The extension of a partial coloring of G can be viewed as a list coloring problem, where the lists on precolored vertices have cardinality one, and the list on any other vertex contains the colors that do not appear on precolored vertices in its neighborhood.In this paper, we study Hall's condition in the context of extensions of partial colorings.
where L φ is the natural list assignment associated with φ: The definition of "Hall k-extendible," was first stated in [5] and was called "Hall kcompletable," due to the relationship that paper explored with completing partial Latin squares.We have adopted the term "extendible" as this is more common in the PrExt literature, and precoloring extensions are the focus of this paper.The following basic results regarding k-precolorings were also established in [5].
The main result of this paper addresses a question asked in [5] that was motivated by Brooks' theorem ([8]): If G is a graph that is neither complete nor an odd cycle, is G Hall ∆(G)-extendible?In Section 2, we provide a fully affirmative answer to this question.This proves a natural precoloring extension version of Brooks' theorem: when precoloring a graph with a palette of ∆(G) available colors, one is guaranteed an extension provided the obvious necessary condition in Definition 2 is not violated.
The following theorems highlight the fact that the concept of Hall extendibility diverges considerably from that of colorability.2 illustrates this; we may precolor the vertices of degree one in the graph G k so as to produce the Hall assignment shown in Figure 1.(We shall refer frequently to the family of graphs in Figure 2 throughout, especially the graph G 3 .)One might conjecture that this is due to the existence of a list of cardinality one on an uncolored vertex, but in fact, bipartite graphs may fail to be Hall 3-extendible, even if all lists of non-precolored vertices have cardinality at least two.Consider the graph in Figure 3.The 3-precoloring is Hall by Theorem 4, but it is not extendible.This behavior is not limited to bipartite graphs.
In Section 3, we further investigate surprising ways in which this graph parameter behaves, looking at how the Hall k-extendibility of G relates to the Hall k-extendibility of its subgraphs.In Section 4, we discuss how increasing the number of colors can affect the Hall extendibility of various graphs.

Every graph G is Hall ∆(G)-extendible
In [13], the authors established the following in response to the question of Bobga et.al [5] on whether any graph G is Hall ∆(G)-extendible: We now completely answer this question in the affirmative.Some preliminaries are required . The graph G is called degree-choosable if G has a proper L-coloring for every degree-assignment L. A graph is a Gallai tree if every block of the graph is either a clique or an odd cycle.
Furthermore, the degree-assignments L under which Gallai trees fail to be proper L-colorable have a restricted form: Theorem 14 (Borodin [6,7], Erdős et.al [9]).If L is a degree-assignment for a connected graph G and there is no L-coloring of G, then where B(v) is the set of blocks containing v, and for each block B, L B is a set of χ(B) − 1 colors.
Lemma 15.If L is the list assignment to Gallai tree G as described in Theorem 14, then (G, L) fails Hall's condition.
Proof.We prove by induction on b, the number of blocks of G, that any list assignment to a Gallai tree satisfying (i) and (iii) from Theorem 14 fails ( * ) on the entire graph G.
Returning to G itself, we have two cases: If σ is any of the m − 1 colors assigned to B, then σ can contribute at most one more to α(G(σ, L)).Thus, In either case, (G, L) fails to satisfy ( * ).
The following observation will be used in the proof of Lemma 17.
Proof.Viewing the precolored vertices V 1 − V 0 as vertices with lists of cardinality 1, we see that Lemma 17.If G is a complete graph or an odd cycle, then any ∆(G)-precoloring to G is not Hall.Thus, all graphs of this type are Hall ∆(G)-extendible by default.
Proof.By the contrapositive of Observation 16, it suffices to show that the empty ∆(G)precoloring, that is, with and each of the two colors contributes exactly k to the right side of ( * ).If G = K m , then each of the m − 1 colors contributes exactly one to the right side of ( * ).In either case, the inequality is violated.
Proof.By Lemma 17 and Brooks' Theorem (see [8]), we may assume χ(G) ∆(G).Suppose that the theorem is false, and let G be a graph of minimum order where the theorem fails.Let φ be a Hall ∆(G)-precoloring of G that is not extendible.Suppose V 0 ⊂ V is the set of vertices that are precolored by φ, and L φ the corresponding Hall list assignment to we conclude by minimality of G that H +U is Hall ∆(H +U )-extendible and thus by Theorem 9 (statement 1), H + U is Hall ∆(G)-extendible.So, any Hall ∆(G)-precoloring of H + U is extendible.Precolor U to match φ and then color H.Because H was arbitrary and no two components of G share any edges, iterating this creates a proper ∆(G)-coloring of G .This is a contradiction to the claim that φ was not extendible.Thus, the Claim is established.
For every v ∈ V (G ), we have that |L φ (v)| deg G (v), i.e., L φ is a degree-assignment to G .By Theorem 14, because G does not have an L φ -coloring, we must have , where B(v) is the set of blocks containing v, and for each block B, L B is a set of χ(B) − 1 colors.By Lemma 15, L φ cannot be Hall, a contradiction because by hypothesis and Theorem 9 (statement 3(b)), the restriction of L φ to G must be a Hall-assignment.Hence, no such counterexample G can exist to the theorem.
Theorem 18 adds substantially to the categories of graphs that are total Hall extendible, including graphs with χ(G) = ∆(G) and graphs with ∆(G) 3. We close this section with a table of graphs (see Table 1) that have been shown to be total Hall extendible.Note that several of these categories that were previously shown to be total Hall extendible, such as the Petersen graph, are included in the category of subcubic graphs, demonstrating the strength of Theorem 18.The graphs in Table 1 can be viewed as graphs on which precolorings with at least χ(G) colors can "easily" be extended.That is, any precoloring with k χ(G) colors that does not violate the obvious necessary condition in Definition 2 can be extended to a proper k-coloring of the graph.We believe further expansion of this list would be an interesting contribution to the PrExt problem.

Hall-Extendibility Under Subgraph Containment
In this section, we briefly discuss how Hall extendibility behaves under subgraph containment.Again, the divergence from colorability is remarkably strong.
G is a k-coloring of H.However, the same behavior is not observed for Hall k-extendibility.For example, consider the graph G in Figure 4. Deleting the vertex v from G results in a subgraph H that can be precolored to obtain the graph with the Hall list assignment shown in Figure 1, so G contains a subgraph that is not Hall 3-extendible.However, the Hall 3-precoloring of G − v (which is not extendible) viewed as a precoloring of G would leave v with an empty list, and therefore is not a Hall 3-precoloring of G. Furthermore, it has been verified through case analysis that G is, in fact, Hall 3-extendible.One might conjecture that this behavior is due to the fact that the precoloring is not Hall on G − H ∼ = K 1 ; however, the following proposition illustrates that is not the case.Theorem 20.Suppose H is not Hall k-extendible, and G is a supergraph of H such that each vertex in G−H has at most t neighbors in H, where t ∈  Therefore we may assume that F contains vertices of H and G − H. Let H F and G F be the subgraphs of F induced on H and G−H, respectively.Let X be a maximum independent set in G F (refer to Figure 6 Let H F be the subgraph induced by V (H F ) − N H F (X) and F be the subgraph induced on the remaining vertices of F .
Since there are no edges between H F and X, and X is an independent set, Since only vertices of H are precolored and any vertex in X is adjacent to at most t vertices in H, we have that |L(x)| k − t for every x ∈ X.However, F itself contains no precolored vertices, so if x ∈ X, then each y ∈ N H F (x) contributes one to |L(x)| as each such y reduces the potential number of precolored vertices to which x can be adjacent.
Recall that if G and H are graphs, then the cartesian product of G and H is the graph G H having vertex set V (G) × V (H) and edge set E(G H) defined as {{(g 1 , h 1 ), (g 2 , h 2 )} : h 1 = h 2 and {g 1 , g 2 } ∈ E(G) or g 1 = g 2 and {h 1 , h 2 } ∈ E(H)}.
The following corollary will be used in Section 4.
Corollary 21.Suppose H is not Hall k-extendible, and G H is k − 1 colorable.Then G H is not Hall k-extendible.
Hall extendibility is poorly behaved under edge deletions.Consider first the case where G is not Hall k-extendible; must G − e maintain this property?Consider the graph H formed by adding an extra neighbor a of degree 1 to vertex u in the graph G k in Figure 2.Both H and H − {ua} fail to be Hall t-extendible for any 3 t k.Hence it is possible to delete an edge from a graph that is not Hall k-extendible and retain this property.However, deleting an edge from the graph G k in Figure 2 connecting vertex u to one of its leaves produces a graph that is Hall k-extendible (since any k-precoloring leaves either a forest induced on the precolored vertices or bipartite graph with a 2-assignment), but is not Hall t-extendible for any 3 t k − 1.Finally, deleting an edge of the 4-cycle results in a tree, and hence a graph that is Hall t extendible for all t 2. Therefore deleting an edge may have a drastic effect on the extendibility of the graph.
Note that if G is not Hall k-extendible, but G − e is for any e ∈ E(G), then G may be considered Hall k-(edge)-critical.An example of such a graph would be G 3 from Figure 2. Deleting any single edge results in a graph that is Hall 3-extendible.
Hall extendibility also behaves unpredictably under edge contraction, as the following shows: Proposition 22.Let G = G 3 be the graph shown in Figure 2 (where k = 3).Although G is not Hall 3-extendible, contracting any edge of G produces a graph that is Hall 3extendible.
Proof.Let e ∈ E(G) and let H be the graph obtained from G by contracting e. Clearly ∆(H) 3.If ∆(H) = 3, then H is Hall 3-extendible by Theorem 18.Thus, we may assume that ∆(H) ∈ {4, 5}.Let C denoted the unique 4-cycle (if e is a bridge in G) or 3-cycle (if e is a non-bridge in G) in H. Take any   Lemma 25.For any Hall assignment to a graph, a vertex can be a mandatory witness to at most one color.
Proof.Let L be a Hall assignment to G and suppose some vertex the electronic journal of combinatorics 23(4) (2016), #P4.19 Proof.Suppose otherwise, that G is Hall k-extendible, but G−u has a Hall k-precoloring φ that is not extendible.Since φ is a valid precoloring of G that clearly cannot be extended, (G, L φ ) does not satisfy Hall's condition, so there must exist a subgraph H of G such that (H, L φ ) does not satisfy ( * ).If u / ∈ V (H), then H is a subgraph of G − u, but then (G − u, L φ ) would not satisfy Hall's condition, a contradiction.Hence, u ∈ V (H).Note (G − u, L φ ) satisfies Hall's condition but (H, L φ ) does not satisfy ( * ).Therefore, if H = H − u, then (H , L φ ) must satisfy ( * ) with equality.Furthermore, u cannot contribute at all to the summation on the right side of ( * ) for H, for otherwise (H, L φ ) would satisfy ( * ).Thus the unique neighbor of u, say v, is not precolored by φ.Hence u has a full list, but v is a mandatory witness for all k colors.As k 2, this violates Lemma 25.
The conclusion of Corollary 27 is not necessarily true if u is a vertex of degree 2. The graph in Figure 4 is Hall 3-extendible, but deleting the edge connecting v to its unique neighbor of degree 2, say u, leaves a graph that is not Hall 3-extendible because the neighbors of the 4-cycle can be 3-colored to produce the lists in Figure 1.However the deletion of the edge uv prevents the list on v from ever being empty, and one can verify that Hall's condition is satisfied with uv removed.
Recall the center of a graph G is the graph C(G) obtained by iteratively deleting vertices of degree one until none exist.

Hall Spectrum for Certain Graph Classes
In light of the fact that Hall k-extendibility does not imply Hall (k + 1)-extendibility, we introduce the following definition.
Definition 29.The Hall spectrum of a graph G is a binary vector h(G) = [h 0 , . . ., h β ] where β = ∆(G) − χ(G) and Theorem 18 implies that h β (G) = 1 for every graph G.Total Hall extendible graphs have a spectrum consisting entirely of 1's.In this section, we discuss the variety one finds in the behavior of the Hall spectrum.
The following result shows that although Theorem 10 implies that h 0 (G) = 1 whenever G is a bipartite graph, this fails for graphs with higher chromatic number.
Proposition 30.For every k 3, there exists a graph G where χ(G) = k but G is not Hall k-extendible.Proof.Let G be the graph obtained from G k shown in Figure 2 by adding an edge between each pair of the k − 1 neighbors of u that have degree one.It is routine to verify this graph is not Hall chromatic extendible.
Theorem 20 implies that many bipartite graphs are not Hall 3-extendible.Before we discuss this further, we prove a helpful lemma that provides a sufficient condition for a precoloring of a bipartite graph to satisfy Hall's condition.
Lemma 31.Suppose G is a bipartite graph.If φ is a k-precoloring of G for k 3 such that |L φ (v)| 2 for all but at most one vertex and no lists are empty, then φ is Hall.
Proof.Assume otherwise, that for some subgraph H of G, (H, L φ ) does not satisfy ( * ).By Theorem 4 we may assume there exists some vertex be the bipartition of V (H), and assume by symmetry that x ∈ H 1 .For each σ ∈ [k], we have α(H(σ, L φ )) α(H i (σ, L φ )) for i ∈ {1, 2}.Since H 1 and H 2 are independent sets, for i ∈ {1, 2} we have (The strict inequality comes from the assumption that (H, L φ ) does not satisfy ( * ).)We consider two cases.
2 .Since all vertices in H 2 have lists of cardinality at least 2, . All vertices in H 1 have lists of cardinality at least two, except for L φ (x).Hence By Theorem 20, any bipartite graph G containing the graph H = G 3 from Figure 2  (where k = 3) as an induced subgraph and having the property that no vertex in G − H has more than one precolored neighbor in H will fail to be Hall 3-extendible.In fact, we can extend this result to bipartite graphs with arbitrarily large girth in the following way.
Theorem 32.Let G be a bipartite graph with girth g 6 and δ(G) k + 1 where k 3.If G contains a cycle C of length g such that C is the only g-cycle containing v for all v ∈ V (C), then G is not Hall k-extendible.
Proof.Let V (C) = {v 0 , v 1 , . . ., v g−1 }.For any v i and v j in V (C), the distance, d C (v i , v j ), along C from v i to v j satisfies d C (v i , v j ) g/2.
Claim: For every v i and v j in V (C), the length of a shortest path from v j to v i whose internal vertices belong to G − C is at least four.
Proof of Claim: Suppose that the length of such a path, call it P , was less than four for some v i , v j in V (C).Let P be a shortest path on C from v i to v j .Then P P is a cycle different from C of length g/2 + 3.As g 6, we have g, contradicting the hypotheses.Thus, the Claim has been established.
Since δ(G) k + 1 and C has no chords (since its length is the girth), v 0 has at least k − 1 neighbors outside C; let Y = {y 1 , y 1 , . . ., y k−1 } be k − 1 such neighbors.For each i 1, let {x i 1 , x i 2 , . . ., x i k−2 } be k − 2 neighbors of v i outside C. By the observation above, note that x i j = x k l for any i, j, k, l.Let and let with i + 1 for all i ∈ [k − 3], and precolor x g−1 k−2 with color k.Finally, precolor x j i with color i for all 2 j g − 2. Figure 7 illustrates H and its precoloring when k = 5.By Lemma 31 the coloring is Hall but not extendible, therefore H is not Hall k-extendible.
By Theorem 20, it suffices to show that every vertex in G−H has at most one neighbor in H.By the observation, the distance between any two vertices in H is at most g/2 + 2. If w ∈ V (G − H) has two neighbors z 1 and z 2 in H, then a shortest path between z 1 and z 2 in C together with the path z 1 , w, z 2 forms a cycle of length at most g/2 + 4. When g 8, g/2 + 4 g, a contradiction.When g = 6, g/2 + 4 is odd, so in fact the cycle has length at most 6, again a contradiction.
Notice that the hypotheses in Theorem 32 are much stronger than needed.Cycles smaller than length g that are sufficiently far from C can be permitted, and only the vertices of C need to satisfy the minimum degree condition.The neighbors of the cycle vertices can overlap, since many of them are precolored the same colors; for example, v 1 , v 2 , . . ., v g−2 could be adjacent to a common set of k − 1 vertices.This illustrates how rare it is for a bipartite graph to be Hall k-extendible for some k δ(G).The behavior of the Hall spectrum beyond coordinate δ(G) − 2 is still poorly understood.
Combining Theorem 9 (statement 1), Theorem 18, and Lemma 31, it is straightforward to show the following: Corollary 33.For all k, 3, the bipartite torus C 2k C 2 is total Hall-extendible.
For some graph families in addition to those that are total Hall extendible, we have completely determined the Hall spectrum.For example: Theorem 34.Let n 6.The Hall spectrum of the hypercube We will prove this theorem through a series of lemmas.
Lemma 35.For every n 6, the graph Q n is not Hall 3-extendible.
Proof.We induct on n.When n = 6, view V (Q 6 ) as the bitstrings of length six.Let φ precolor 100100 and 110010 with color 1, 100010 and 000100 with color 2, and 010001 with color 3.It is routine to verify that L φ restricted to the 4-cycle (000000, 100000, 110000, 010000) has lists matching Figure 1 and hence φ is not extendible.Furthermore, vertex 100000 is the only vertex with a list of cardinality less than two, so Lemma 31 implies that Q 6 is not Hall 3-extendible.Now assume Q n is not Hall 3-extendible for some n 6.By Corollary 21, It is unknown whether Q 4 or Q 5 are Hall 3-extendible.However, the next lemma shows that Q 5 is not Hall 4-extendible.
Lemma 36.For every n 5, the graph Q n is not Hall (n − 1)-extendible.
Throughout the proof we shall construct new bitstrings via concatenating X n or Y n on the right.For each n 5, we recursively define an (n − 1)-precoloring φ n of Q n that has the following three properties, which we shall refer to in the remainder of the proof: (2) There are no empty lists and 1000X n−4 is the only vertex with a list of cardinality one.
( We shall now define a precoloring φ n of Q n as follows: Case 1: v / ∈ S. If v ends in a 1, then φ n does not precolor v.If v ends in a 0, v = v 0, where v ∈ V (Q n−1 ).Then φ n precolors v if and only if φ n−1 precolors v , in which case φ n (v) = φ n−1 (v ).That is, φ n restricted to the subgraph K ∼ = Q n−1 of Q n consisting of all vertices ending in a 0 is the same as φ n−1 for vertices not in S. We verify that φ n satisfies the necessary properties (again refer to Figure 8).It is clear that property (3) is satisfied, since exactly two vertices with a 1 in the final position are colored.Property (1) is maintained since the four vertices given color n − 1 dominate the 4-cycle (00000X n−5 , 10000X n−5 , 11000X n−5 , 01000X n−5 ).Finally, property (2) is maintained since at each iteration, all additional precolored vertices are given color n − 1, and hence ∅ = L φ n−1 (v ) ⊆ L φn (v) for each vertex v / ∈ S. Finally for any fixed n, the (n−1)-precoloring φ n is Hall because property (2) holds, yet it is not extendible because property (1) holds.Thus Q n is not Hall (n−1)-extendible.
Theorem 34 now follows easily from these lemmas.
Proof of Theorem 34.Let n 6.The Hall spectrum h(Q n ) has length β + 1 = n − 1.Since Q n is bipartite, it is clear that h 0 (Q n ) = 1.By Theorem 18, h n−2 (Q n ) = 1.It remains to verify that h k (Q n ) = 0 for all k ∈ {1, . . ., n − 3}.By Lemmas 35 and 36, h 1 (Q n ) = h n−3 (Q n ) = 0. Now consider Q n for some n > 5 and 4 k n − 1; we wish to show that Q n is not Hall k-extendible.By Lemma 36, Q k+1 is not Hall k-extendible, and k + 1 < n.As in the proof of Lemma 36, applying Corollary 21 inductively shows that Q n is not Hall k-extendible.
The Hall spectrum of wheel graphs with even cycles differs from any other graphs discussed so far.The wheel graph W n is defined as an n-cycle v 1 , v 2 , . . ., v n with a dominating vertex v 0 .
Theorem 37. Let n 10.The wheel graph W n is Hall k-extendible for all k 4. Furthermore, if n is even, then W n is not Hall 3-extendible.There are many similar questions that could be asked regarding edge and vertex deletion.
Most importantly, as total Hall extendible graphs can be viewed as graphs in which precolorings with at least χ(G) can easily be extended (provided the obviously necessary Hall's condition is satisfied), perhaps the most important question to answer would be the following: Question 3: Does there exist a characterization of total Hall extendible graphs?

Figure 2 :
Figure 2: A family of graphs G = {G k : k 3} that are Hall 2-extendible because they are bipartite, but for any k 3, the graph G k is not Hall k-extendible.

Figure
Figure2illustrates this; we may precolor the vertices of degree one in the graph G k so as to produce the Hall assignment shown in Figure1.(We shall refer frequently to the family of graphs in Figure2throughout, especially the graph G 3 .)One might conjecture that this is due to the existence of a list of cardinality one on an uncolored vertex, but in fact, bipartite graphs may fail to be Hall 3-extendible, even if all lists of non-precolored vertices have cardinality at least two.Consider the graph in Figure3.The 3-precoloring is Hall by Theorem 4, but it is not extendible.This behavior is not limited to bipartite graphs.

If b = 1 ,
then G is an odd cycle or a clique.It is routine to check that Hall's condition fails on G, establishing a basis.Now suppose that B is a leaf block of G (a block with exactly one cut-vertex in G) and v is the unique cut-vertex in B. Delete all vertices in B except v, and remove the colors in L(v) that are assigned to B. Let L be the resulting list assignment to G = G − (B − v).Note that properties (i) and (iii) of the hypothesis still hold for G .By the induction hypothesis, (G , L ) fails ( * ) and so v

Proposition 19 .
For every k 3, there exists a graph G such that G contains a subgraph H with a Hall k-precoloring φ : V 0 → [k] that is not extendible and (G − H, L φ ) satisfies Hall's condition, but (G, L φ ) does not satisfy Hall's condition.Proof.Let k 3. We claim the graph G = J k shown in Figure5satisfies the proposition.The graph H = G − v is a subgraph of G that has a Hall k-precoloring φ= φ k (which is not extendible) shown in Figure5; note that the lists on a, b, c, and d correspond to the list assignment shown in Figure1.Furthermore, because V (G − H) = {v} and L(v) = {1, 2, 3}, we have that (G − H, L φ ) trivially satisfies Hall's condition.However, it is routine to verify that Hall's condition fails on the subgraph induced by {a, b, c, d, v}.Therefore, (G, L φ ) does not satisfy Hall's condition.It is obvious that if one or more of the components of G are not Hall k-extendible, then G itself is not Hall k-extendible.The next theorem indicates a nontrivial sufficient condition for G to inherit a Hall k-precoloring from a subgraph H that is not extendible.

Figure 5 :
Figure 5: A family of graphs J = {J k : k 3} shown with a non-extendible k-precoloring φ k from Proposition 19.The shaded triangular region indicates that the vertex v ∈ V (J k ) is adjacent to the k − 3 vertices colored 4 through k when k 4; when k = 3, the vertex v ∈ V (J 3 ) is adjacent only to the vertices a, b, c, d.

Figure 6 :
Figure 6: Illustration for the proof of Theorem 20.All precolored vertices of G must lie in H −H F (dotted region).Here, N(X) = N H F (X), A x = N H F (x) and B x = N H (x)−A x .By hypothesis, |A x | + |B x | t.
thus H has Hall number 1 by Theorem 7, and since L φ | H is a Hall 1-assignment, φ is extendible.Therefore, we shall assume that V 0 ∩ V (C ) = ∅, i.e., φ precolors only vertices of degree 1 in H. Clearly if φ extends to a proper 3-coloring of C , then φ also extends to a proper 3-coloring of H since every vertex of v ∈ V (H ) \ V (C ) has only one neighbor in V (C ).Thus we need only verify that φ extends to a proper 3-coloring of C .If C is a 3-cycle, then L φ restricted to C is a Hall 1-assignment of a clique, which has Hall number 1 by Theorem 7, and φ extends.It remains to consider C a 4-cycle.Let V (C ) = {a, b, c, d} and suppose that a is the unique vertex on C which had two neighbors of degree 1 in G: {a 1 , a 2 }.If at most one of a 1 and a 2 are precolored, if φ(a 1 ) = φ(a 2 ), or if one of the edges aa 1 or aa 2 was contracted, then L φ | C is a Hall 2-assignment to C , which is bipartite, and thus φ extends to a proper 3-coloring of C .If both are precolored and φ(a 1 ) = φ(a 2 ), then since we are guaranteed that C has at least one vertex among {b, c, d} with a full list and the other two vertices have lists of cardinality two, we can color a with[3] \ {φ(a 1 ), φ(a 2 )} and are guaranteed a way to extend φ.Definition 23.Let L be a Hall assignment to a graph G and let σ ∈ C. A vertex v ∈ G(σ, L) is called a mandatory witness to color σ for the list L if the list assignment L created from L by removing σ from L(v) is not a Hall assignment to G. Observation 24.If any vertex is a mandatory witness to a color in a Hall assignment L to G, then |V (G)| = σ∈C α(G(σ, L)).

Figure 7 :
Figure 7: A Hall 5-precoloring of the graph H from the proof of Theorem 32 which is not extendible.

Question 2 :
the electronic journal of combinatorics 23(4) (2016), #P4.19If G is Hall k-extendible, then must there exist an edge e ∈ E(G) such that G − e is also Hall k-extendible?Note that Corollary 28 implies that any Hall k-extendible graph G in which G − e fails to be Hall k-extendible for every e satisfies C(G) = G.

Table 1 :
A table of some graph families that are total Hall extendible.
We claim that H is not Hall k-extendible.Precolory i with color i + 1 for each i ∈ [k − 1].Precolor x 1 i with color i + 1 for each i ∈ [k − 2].Precolor x g−1 i