Brooks' theorem with forbidden colors

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete graph and $P \subseteq V(G)$ is a set of vertices where either (i) at most $\Delta(G)-2$ colors are forbidden for every vertex in $P$, and any two vertices of $P$ are at distance at least $4$, or (ii) at most $\Delta(G)-3$ colors are forbidden for every vertex in $P$, and any two vertices of $P$ are at distance at least $3$, then there is a proper $\Delta(G)$-coloring of $G$ respecting these constraints. In fact, we shall prove that these results hold in the more general setting of list colorings. These results are sharp.


Introduction
Brooks' classic theorem on graph coloring states that if G is a connected graph, not isomorphic to a complete graph or an odd cycle, then G is properly ∆(G)-colorable, where ∆(G) as usual denotes the maximum degree of G.This theorem has been strengthened in many different ways, see e.g. the recent survey [8].In particular, a list coloring version of Brooks' theorem was obtained already in the seminal paper on list coloring by Erdős et al [10], and also independently by Vizing [12].
A precoloring (or partial coloring) of a graph G is a proper coloring of some subset V ′ ⊆ V (G).Given a precoloring ϕ of G, we are usually interested in finding an extension of ϕ, that is, a proper coloring of G that agrees with the partial coloring ϕ.If there is such a coloring (using the same number of colors as ϕ), then ϕ is extendable.
A precoloring extension version of Brooks' theorem was obtained by Albertson et al [2] and, independently, by Axenovich [3]: if P is an independent set, ∆(G) ≥ 3 and the minimum distance between any two vertices in P is 8, then any proper coloring of P can be extended to a proper ∆(G)coloring of G, unless G contains a copy of K ∆(G)+1 , where K n as usual denotes a complete graph on n vertices.In general, the condition on the distance is tight, but Voigt obtained improvements for the case when G is 2-connected [13,14].
In this short note we consider the similar problem of constructing proper ∆(G)-colorings avoiding certain colors; that is, given a subset P ⊆ V (G) of the vertices of a graph G where every vertex of P is assigned a set of forbidden colors from {1, . . ., ∆(G)}, we are interested in finding a proper ∆(G)coloring of G respecting these constraints.These type of questions go back to a paper by Häggkvist [11] and although they arise naturally in problems where a coloring is constructed sequentially, it seems that they so far primarily have been studied in the setting of edge colorings, see e.g.[5,6,7,9] and references therein.In particular, a variant of Vizing's edge coloring theorem with forbidden colors was obtained in [9].Note that it is not possible to prove a variant of König's edge coloring theorem with forbidden colors (even if only one forbidden color is assigned to edges in a matching), as the coloring of the graph in Figure 1 shows; here K (i) n,n −e denotes a copy of the complete bipartite graph K n,n where an arbitrary edge has been removed (which is dashed in the figure).Since color 1 cannot be used on any of the edges at the top of the figure, it cannot be used on any edge incident with the bottom vertex u either.Hence, there is no proper ∆(G)-edge coloring respecting the forbidden colors.
Note that here we can make the distance between the colored edges arbitrarily large by adding more copies of K n,n − e along with connecting edges between different copies of K n,n − e. u 1 1 1 n,n − e K  Let us note that the general problem of coloring a given graph properly subject to the condition that some colors cannot be used on certain vertices is certainly NP-complete since the general vertex (list) coloring problem is.
Here we prove that if G is a graph with ∆(G) ≥ 4, not containing a copy of K ∆(G)+1 , and P ⊆ V (G) an independent set where (i) at most ∆(G) − 2 colors are forbidden for every vertex in P , and any two vertices of P are at distance at least 4, or (ii) at most ∆(G) − 3 colors are forbidden for every vertex in P , and any two vertices of P are at distance at least 3, then there is a proper ∆(G)-coloring of G respecting these constraints.It remains an open problem whether (i) holds in the case when ∆(G) = 3.
Comparing this to the aforementioned results on precoloring extension [1,2], we see that allowing just the slight extra flexibility of having one or two extra colors on the vertices in P implies that the distance condition can be relaxed significantly.
Moreover, as for the results on precoloring extension in [1,2], we shall prove that these results hold in the more general setting of list coloring of graphs.This is proved in the next section, where we also give examples showing that they are sharp.In Section 3 we briefly consider the related question of avoiding a given partial coloring ϕ, i.e. the problem of finding a coloring f of a graph which differs from ϕ on every vertex that is colored under ϕ, and give some observations and remarks on this problem.

Main Result
Before proving our main result, let us introduce some terminology.The block-cutpoint graph of a graph G has a vertex for each block of G and a vertex for each cut-vertex of G, where a cut-vertex v is adjacent to a block B if v ∈ V (B).A leaf block in a graph G is a block of G that contains at most one cut-vertex.We shall use standard graph theory notation; in particular, d G (v) denotes the degree of a vertex v in G and an induced subgraph of G is denoted by G[S], where S is a set of vertices or a set of edges.
Given a graph G, assign to each vertex v of G a set L(v) of colors (positive integers).Such an assignment L is called a list assignment for G and the sets L(v) are referred to as lists or color lists.If all lists have equal size k, then L is called a k-list assignment.Usually, one is interested in finding a proper vertex coloring ϕ of G, such that ϕ(v) ∈ L(v) for all v ∈ V (G).If such a coloring ϕ exists then G is L-colorable and ϕ is called an L-coloring; if L is clear from the context, then we just call ϕ a list coloring and say that G is list colorable when the coloring exists.Furthermore, G is called k-choosable if it is L-colorable for every k-list assignment L. The least number k such that G is k-choosable is called the list-chromatic number (or choice number) of G and is denoted by χ l (G).
As mentioned above, Brooks' theorem holds in the setting of list coloring; that is, if G is a connected graph which is not isomorphic to an odd cycle or a complete graph, then χ l (G) ≤ ∆(G).As in the aforementioned papers on precoloring extension, we shall use the following stronger theorem proved in [4,10].A Gallai tree is a connected graph in which every block is a complete graph or an odd cycle.A graph G is degree-choosable if it has an L-coloring whenever L is a list assignment such that , 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.
Note that this implies (as remarked in [2]) that each block B is an |L B |-regular graph, and that all vertices of a single block that are not cut-vertices of G have the same list.Furthermore, we shall need the following.
Corollary 2.2.Suppose that T is a Gallai tree, L a supervalent list assignment satisfying conditions (a), (b), and (c) of Theorem 2.1, and that there is a leaf block B L of T with ∆(T ) ≥ 3 vertices u 1 , . . ., u ∆(T )−1 , v, where v is a cut-vertex of T of degree ∆(T ).Assume further that u 0 is a vertex of degree ∆(T ) − 1, not contained in B L , and that there are colors c 1 , c 2 , c 3 such that ∈ L(u i ), for i = 0, . . ., ∆(T ) − 1, and Set U = {u 0 , u 1 , . . ., u ∆(T )−1 } and define the list assignments L ′ and L ′′ for T by setting The proof of this corollary is a rather straightforward application of Theorem 2.1; we briefly sketch an argument.
The conditions imply that there must be a block , where we use the notation from Theorem 2.1.Thus there must be a block B 3 in T containing x, such that c 2 ∈ L B 3 and c 1 ∈ L ′ B 3 .Now, since L and L ′ only differs on the lists for the vertices in U , B 3 ∼ = K 2 , and, more generally, there is path from v to u 0 in T , all edges of which lie in blocks that are isomorphic to K 2 .Furthermore, since T is not L-or L ′ -colorable, every block B i of this path satisfies that since T is a Gallai tree, this is the only path from v to u 0 in T .Using this property, it is now easy to check that this implies that T must be L ′′ -colorable.
The following is the main result of this paper.
Theorem 2.3.Let G be a connected graph with maximum degree ∆(G) ≥ 4 which is not complete and Before proving the theorem, let us demonstrate that it is sharp both with respect to list sizes and the distance conditions.For the case ∆(G) = 3, we do not know whether a result as in (i) holds, but the example below shows that distance 4 would be best possible.
That the list size cannot be improved in part (i) follows by taking two copies K ∆−1 and K ∆−1 of K ∆−1 , where all vertices have the list {1, . . ., ∆} and making exactly one vertex of K ∆−1 are adjacent to a vertex u with the list {1} and all vertices of K (2) ∆−1 are adjacent to a vertex v = u with the list {1}.Clearly, this graph is not list colorable.
That the distance condition in part (i) is best possible follows from the fact that the ∆-regular graph in Figure 2 is not list colorable.Here all vertices that are contained in copies of K ∆ are assigned the list {1, . . ., ∆}, while the vertices not contained in the copies of K ∆ are assigned the list {1, 2}.Since at least two of the vertices not contained in the copies of K ∆ must be colored by the same color, this graph is not list colorable.Hence, the distance condition in part (i) cannot be improved.Moreover, this example also shows that the list size in part (ii) is sharp.
That the distance condition in part (ii) is best possible can be seen by taking a copy of K ∆−1 along with two vertices u and v that are joined to every vertex of K ∆−1 by an edge.Assign the list {1, 2, 3} to u and the list {4, 5, 6} to v and the list {1, 2, . . ., ∆} to all vertices of K ∆−1 .Then this graph is not list colorable.Let us now turn to the proof of Theorem 2.3.We shall need the following lemma.Recall that an acyclic subgraph of a digraph is a subgraph with no directed cycle.Lemma 2.4.If D is a digraph and S ⊆ V (D) a set of vertices v satisfying that the outdegree is strictly larger than the indegree of v, then there is an ayclic subgraph D ′ of D such that every vertex of S has outdegree 1 in D ′ .
Since every vertex of S has strictly larger outdegree than indegree, this is straightforward; we leave the details to the reader.
Proof of Theorem 2.3.Since (in both cases) P is independent and 1 that any component of H that is not L ′ -colorable is a Gallai tree.Moreover, since any vertex in G is adjacent to at most one vertex of P , every component of H that is not L-colorable has minimum degree ∆(G) − 1, and all vertices of such a component has degree ∆(G) in G.
Suppose that some component T of H together with the restriction of L ′ to T is not list colorable, and thus satisfies the conditions in Theorem 2.1.We call such a component ϕ ′ -bad (or just bad).Suppose further that some vertex p ∈ P is adjacent to exactly ∆(G) − 1 vertices of a leaf block B 1 ∼ = K ∆(G) in T and has no other neighbor in T .By recoloring p with another color of its list we get a new coloring ϕ ′′ of P so that T is not ϕ ′′ -bad.Similarly, if p ∈ P has only one neighbor in T , then by recoloring p we obtain a coloring ϕ ′′ such that T is not ϕ ′′ -bad.Our proofs of both part (i) and (ii) relies heavily on this simple observation.
Without loss of generality, we shall assume that every component of H satisfies the conditions (a) and (b) of Theorem 2.1 (with L ′ in place of L); otherwise, we just consider a suitable subgraph of H, since components not satisfying these conditions cannot be bad.
After these initial observations, let us now prove part (i) of the theorem.Since G is not complete, and any two vertices of P are at distance at least 4, no component of H is isomorphic to a complete graph, and no two vertices of P are adjacent to vertices of the same block in H, and thus no vertex of P is adjacent to vertices of two different leaf blocks.This implies that every component of H contains at least two leaf blocks.Moreover, if T is such a component, then every leaf block B L of T is isomorphic to K ∆(G) , ∆(G) − 1 vertices of B L are adjacent to the same vertex of P , and the cut-vertex of B L is not adjacent to any vertex of P .Now, suppose ϕ ′ is a coloring of P so that some component T of H is ϕ ′ -bad.We shall prove that by recoloring some vertices P ′ ⊆ P satisfying that if p ∈ P ′ then p has ∆(G) − 1 neighbors in a leaf block, we can obtain an L-coloring ψ of P so that no component of H is ψ-bad.Indeed, in the following we shall describe a recoloring procedure.
To this end, we shall prove the following claim.Let P 1 be the set of all vertices of P that have at least one neighbor in a leaf block of H, and let V 1 be the set of all vertices in H that are not contained in a leaf-block and has a neighbor in P 1 .

Claim 1. If T is a component in H, then the number of leaf-blocks in T is greater than the number of vertices in
Proof.Let us first note that no block of H contains two vertices of V 1 , as follows from the definition of V 1 .Suppose now that v is a vertex of V 1 .Consider the block-cutpoint graph W of T .Since v has degree at least four in G, and no other vertex of a block containing v has a neighbor in V 1 , it follows that v either is a cut-vertex of degree at least 3 in W , or that there is a block vertex of degree at least 3 in W , whose corresponding block in T contains v.Moreover, since every block of T contains at most one vertex from V 1 , we get that for each vertex of V 1 , there is a unique vertex of degree at least 3 in W . Now, since W is a tree, it contains more leafs than vertices of degree at least three.Hence, the number of leaf blocks of T is greater than We now define a directed graph D in the following way.Each component T in H is represented by a vertex t, and there is a directed edge from a vertex t to another vertex t ′ if there is a vertex p adjacent to ∆(G) − 1 vertices of a leaf block in T that is also adjacent to a vertex of T ′ .Moreover, for every leaf-block of a component T of H for which the adjacent vertex p ∈ P is not adjacent to any other vertex in H, we add a new vertex x p and an arc (t, x p ); thus x p has indegree 1 and outdegree 0 in the digraph D. Furthermore, every directed edge of D corresponds to a vertex in P .
By Claim 1 every vertex of D corresponding to a component of H has larger outdegree than indegree, so it follows from Lemma 2.4 that there is an acyclic directed subgraph D ′ of D such that every vertex corresponding to a component of H has outdegree 1; denote these vertices by S.
Since D ′ is acyclic there is a linear order ≺ of the vertices of S such that if (t, t ′ ) is a directed edge of D ′ , then t ≺ t ′ .
• If t 1 corresponds to a ϕ ′ -bad component and (t 1 , x) ∈ A(D ′ ), then we recolor the vertex p 1 corresponding to the arc (t 1 , x) ∈ A(D ′ ) to obtain a new coloring ϕ 1 of P .Note that by the construction of D ′ and the order ≺, the only component of H that might be ϕ 1 -bad but not ϕ ′ -bad is a component of H corresponding to x, if there is such a component.Moreover, after this recoloring, T 1 is not ϕ 1 -bad.
• If t 1 is not bad, then we just set ϕ 1 = ϕ ′ and continue with the next vertex according to the order ≺.
We continue this process for all vertices of S according to the linear order ≺ and define Lcolorings ϕ 1 , ϕ 2 , . . . .Then every vertex in S is considered once, and the process terminates after a finite number of steps.Now, since t i ≺ t j precisely when there is no directed path from t j to t i in D ′ , it follows that no component T i corresponding to a vertex t i ∈ S can be ϕ j -bad for j ≥ i.Hence, after this process terminates when all vertices of S have been considered, we obtain a proper L-coloring ϕ n where no component of H is ϕ n -bad.This proves part (i).
Let us now prove part (ii).We shall construct a subgraph F ′′ of G for coloring the vertices of P from their lists L.
Consider the bipartite multigraph J with parts P and B, where B contains a vertex for every leaf-block of H, and where a vertex p of P is joined by an edge to a vertex b ∈ B for each edge between p and the block corresponding to b in H.In J, every vertex of P has degree at most ∆(G), and every vertex of B has degree at least ∆(G) − 1 and at most ∆(G).Thus, by König's edge coloring theorem, there is a proper ∆(G)-edge coloring of J.
We denote by J ′ the subgraph of J induced by the edges colored 1, 2, 3, 4. Then ∆(J ′ ) ≤ 4 and every vertex of B has degree at least 3 in J ′ .Note further that if some vertex b ∈ B has degree 3 in J ′ , then b corresponds to a leaf block of H that is contained in a component with at least two leaf blocks.
From J ′ we form a new graph J ′′ with vertex set V (J ′ ) by for every component T of H, where every leaf-block of T corresponds to a vertex of degree 3 in J ′ , arbitrarily selecting two vertices b 1 and b 2 , corresponding to different leaf-blocks of T , and redistributing the edges incident with b 1 and b 2 so that one edge incident with b 2 is "moved to" b 1 ; thus b 1 gets degree four in J ′′ and b 2 gets degree two.Note that this does not change the degrees of vertices in P .Now, by arbitrarily adding edges between pairs of vertices of odd degree in J ′′ , we obtain a graph K where all vertices have even degree at most 4. Thus every component of K is Eulerian and by taking every second edge in an Eulerian circuit of every component of K we obtain a subgraph F K where every vertex has degree 0, 1 or 2. Denote the restriction of this subgraph to J ′′ by F J , and let F be the subgraph of J ′ corresponding to F J .Then every vertex of F has degree at most two and, moreover, every component of H has a leaf block that corresponds to a vertex of degree 2 in F . 1 the set of vertices in B 1 that correspond to leaf blocks in H with at least two different neighbors in P in G. Let P ′ be the set of vertices in P that are adjacent to vertices of B ′ 1 in F .From F ′ we form a new graph F ′′ with V (F ′′ ) = P and where the edge set is constructed iteratively in the following way: • firstly, two vertices of P are adjacent if they have a common neighbor in F ′ ; • secondly, for all vertices p ′ of P ′ we do the following: suppose p ′ is adjacent to b in F .We add an edge between p ′ and another arbitrary vertex p ′′ ∈ P that in J is adjacent to b.
We note the following properties of F ′′ .
(a) Since every vertex of F has degree at most 2, the same holds for the induced subgraph F ′′ [P \ P ′ ].Thus, by construction, every component of F ′′ is unicyclic, i.e., has at most one cycle.
(b) Every component T of H that is isomorphic to K ∆(G) satisfies that two vertices in P with neighbors in T are adjacent in F ′′ .
(c) If the vertices of a leaf block B of H, which corresponds to a vertex in B 2 , has at least two different vertices in P as neighbors in G, then there are two vertices p, p ′ ∈ P , both with at least one neighbor in B, that are adjacent in F ′′ .
We now color the vertices of P as follows.By (a), the core of F ′′ , obtained by successively removing vertices of degree 1 from F ′′ , has maximum degree at most 2. Hence F ′′ is properly colorable from the lists L of vertices in P .We take such a coloring ϕ of P with a minimum number of ϕ-bad components in H. Now, consider this coloring of the vertices of P in the graph G.It follows from Theorem 2.1 that if a component T of H is ϕ-bad, then for every leaf-block B of T , every vertex of P that has a neighbor in B has the same color under ϕ.Now, since every component of H has a leaf block that corresponds to a vertex of B 2 , it follows from (c) that there is a leaf-block B L in T where ∆(G) − 1 vertices are adjacent to the same vertex p in P ; that is, p has at most one neighbor outside B L .
Suppose ϕ(p) = c 1 and L(p) = {c 1 , c 2 , c 3 }.Since p has at most one neighbor outside B L , it follows that by recoloring p by the color c 2 we get a new L-coloring ϕ ′ of P such that either T is ϕ ′ -bad (and then the neighbor of p not in B L is in T ), or T is not ϕ ′ -bad but some other component is ϕ ′ -bad but not ϕ-bad, since ϕ is an L-coloring with a minimum number of ϕ-bad components.Now, since coloring p by the color c 2 does not decrease the number of bad components and p has at most one neighbor outside B L , we may color p by the color c 3 to obtain a coloring ϕ ′′ with fewer bad components.In the case when p has a neighbor outside T , this follows from our initial observation before the proof of part (i), and in the case when p has all neighbors in T , then it follows from Corollary 2.2.In both cases, this contradicts the choice of ϕ and thus completes the proof of the theorem.
It is natural to ask if it is possible to prove a version of Theorem 2.3 where the set P of vertices with shorter lists is allowed to be any independent set.Problem 2.5.Let G be a graph with maximum degree ∆ = ∆(G) ≥ 4, not containing a copy of K ∆+1 , and P an independent set.Is there a k = k(∆) < ∆ such that if L is a list assignment where Since P is independent, it follows from Theorem 2.1 that any minimal counterexample G to this question must satisfy that all vertices of V (G) \ P has degree ∆(G).
In particular, we are interested in whether Problem 2.5 has a positive solution in the case when then lists of P have size |L(v)| = ∆(G) − 1; in the next section we investigate this question further for the special case when all lists are chosen from the set {1, . . ., ∆(G)}.

Avoiding colorings
In this section we consider the problem of coloring a graph G using colors 1, . . ., ∆(G), subject to the condition that some colors cannot be used on certain vertices.Let us begin by noting the following reformulation of Theorem 2.3, which is a generalization of Brooks' theorem for ordinary graph coloring for the case of ∆(G) ≥ 4.An interesting special case of Problem 2.5 is when the lists of P have size |L(v)| = ∆(G) − 1 and all lists use colors from {1, . . ., ∆(G)}, which would be a useful generalization of Brooks' theorem along the lines of Corollary 3.1.We note the following partial result towards such a general theorem.Recall that a partial coloring ϕ of G is avoidable if there is a proper coloring f of G such that f (v) = ϕ(v) for any vertex that is colored under ϕ.Proposition 3.2.Let G be a connected graph with maximum degree ∆(G) ≥ 4 that is not complete, and ϕ a precoloring of a subset P ⊆ V (G).If P is independent and every vertex in V (G) \ P is either adjacent to at least three vertices from P or to no vertex from P , then ϕ is avoidable.This proposition is a special case of a more general result proved below, so we postpone its proof.Conversely, for the case when each vertex in V (G) \ P has a bounded number of neighbors in P , then we have the following.Proposition 3.3.Let G be a connected graph with maximum degree ∆(G) ≥ 4 that is not complete, and ϕ a precoloring of an independent subset P ⊆ V (G).Furthermore, suppose that every vertex in V (G) \ P has at most d 0 neighbors in P , and that every vertex in P has neighbors in at most d 1 components of G − P .If d 0 d 1 < ∆(G) − 1 and every component of G − P contains a leaf block with at least d 0 + 1 distinct neighbors in P , then then there is a proper ∆(G)-coloring of G that avoids ϕ.
Proof.The proof is based on Theorem 2.1 and uses similar ideas as the proof of Theorem 2.3, so we only sketch the argument.The basic idea is to find a proper coloring f of the vertices of P that avoids ϕ, and such that no component of H = G − P together with the list assignment for the vertices of V (G) \ P obtained by setting If T is a component of H so that V (T ) is not L f -colorable for some coloring f of P , then every leaf-block of T is a complete graph K where all vertices have degree ∆(G) in G, and which satisfies that L f (u) = L f (v) for any two vertices u and v of K that are not cut-vertices.Moreover, the cut-vertex w of K satisfies that L f (u) ⊆ L f (w), where u ∈ V (K).Hence, it suffices to prove that there is a coloring f of P so that every component of G − P has a leaf block that does not satisfy this condition.
To this end, we define a "conflict graph" D with vertex set P in the following way.Suppose that K is a leaf block in H that is isomorphic to K k , for some integer k, and where the vertices in K altogether have at least d 0 + 1 distinct neighbors in P .We arbitrarily pick d 0 + 1 vertices in P , each of which has a neighbor in K, and form a clique on those d 0 + 1 vertices.Now, by assumption every component of G − P contains such a leaf block as described in the preceding paragraph, so by repeating this process for one such leaf block of every component of H, we obtain the conflict graph D. The maximum degree of this graph is at most d 1 d 0 , since every vertex of P has neighbors in at most d 1 components of H. Thus, by the list coloring version of Brooks' theorem, there is a proper ∆(G)-coloring f of D that avoids ϕ.Now, by the construction of D every component of H has a leaf block with neighbors in P of d 0 + 1 different colors under f .Since every vertex of V (G) \ P has at most d 0 neighbors in P , this implies that no component of H together with the list assignment L f (v) = {1, . . ., ∆(G)} \ {f (u) : u is adjacent to v} satisfies the conditions in Theorem 2.1 and, consequently, there is a proper ∆(G)-coloring of G that avoids ϕ.
More generally, we might ask what can be said in the case when P is not required to be an independent set?A ∆(G)-coloring of G where every vertex is assigned the same color is not avoidable if χ(G) = ∆(G).We note the following reformulation of a result for edge colorings from [7].Proposition 3.4.If ϕ is a partial k-coloring of a connected graph G that is not complete, where every color appears on at most ∆(G) − k vertices, then there is a proper ∆(G)-coloring of G that avoids ϕ.
The proof is virtually identical to the corresponding result for edge colorings in [7], so we omit it.Note, in particular, that Proposition 3.4 implies that every partial coloring with at most ∆(G) − 1 colored vertices is avoidable, which is sharp by the example of the join of K d with two vertices u and v, where u and all vertices of K d are colored 1.
For the case when the vertices in P induce a subgraph of small maximum degree, we have the following.Proposition 3.5.Let G be a connected graph with maximum degree ∆(G) ≥ 3 that is not complete, and ϕ a precoloring of a subset P ⊆ V (G), where every vertex in V (G) \ P either has at least d neighbors in P or no neighbors in P .If ∆(G[P ]) < d − 2, then there is a proper coloring of G avoiding ϕ.
Proof.Again, the argument is similar to the one in the proof of Theorem 2.3, so we only sketch the proof.
Since every vertex of V (G) \ P has at least d neighbors in P or no neighbor in P , no component T of the graph G − P along with the restriction of L f to T satisfies the conditions in Theorem 2.1.Hence, there is an L f -coloring of G − P .This L-coloring taken together with f is the required coloring.
Proof.Let f be any proper k-coloring of G.For any vertex v that has the same color under f and ϕ we recolor v by the color k + 1; the resulting coloring is proper and avoids ϕ.

Figure 1 :
Figure 1: A bipartite graph G with no proper ∆(G)-edge coloring respecting the forbidden colors.
and any two vertices of P are at distance at least 4, or (ii) |L(v)| ≥ 3 and any two vertices of P are at distance at least 3.

Figure 2 :
Figure 2: A graph with a list assignment and no list coloring.

Next, let B 2
be the set of vertices in B that have degree 2 in F and consider the vertex-induced subgraph F [B 2 ∪ P ].We define B d to be the subset of B 2 containing all vertices that are incident to two different vertices of P in F [B 2 ∪ P ].Let F ′ be the subgraph of F induced by B d ∪ P .Now, in F [B 2 ∪ P ] every vertex of B 2 \ B d is incident with two edges with the same endpoint in P , so every vertex of P with a neighbor in B 2 \ B d , has no other neighbor in F [B 2 ∪ P ].Set B 1 = B 2 \ B d , and denote by B ′

Corollary 3 . 1 .
Let G be a connected graph with maximum degree ∆(G) ≥ 4 that is not complete.Suppose that a set P of vertices is assigned a list of forbidden colors.If either (i) each vertex of P has most ∆(G) − 2 forbidden colors and any two vertices of P are at distance at least 4, or (ii) each vertex of P has most ∆(G) − 3 forbidden colors and any two vertices of P are at distance at least 3, then there is a proper ∆(G)-coloring of G which respects the forbidden colors.