A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs

A $b$-coloring of a graph $G$ is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The $b$-chromatic number of a graph $G$, denoted by $\chi_b(G)$, is the maximum number $k$ such that $G$ admits a $b$-coloring with $k$ colors. We consider the complexity of the problem of deciding whether a graph $G$ has a $b$-coloring with $k$ colors, whenever the value of $k$ is close to one of two upper bounds on $\chi_b(G)$: The maximum degree $\Delta(G)$ plus one, and the $m$-degree, denoted by $m(G)$, which is defined as the maximum number $i$ such that $G$ has $i$ vertices of degree at least $i-1$. We obtain a dichotomy result stating that for fixed $k \in \{\Delta(G) + 1 - p, m(G) - p\}$, the problem is polynomial-time solvable whenever $p \in \{0, 1\}$ and, even when $k = 3$, it is NP-complete whenever $p \ge 2$. We furthermore give an FPT-algorithm parameterized by $k + \ell$, where $\ell$ denotes the number of vertices of degree at least $k$, while we observe that the problem is NP-complete whenever $k$ is unbounded and $\ell = 0$.


Introduction
Given a set of colors, a proper coloring of a graph is an assignment of a color to each of its vertices in such a way that no pair of adjacent vertices receive the same color. In the deeply studied Graph Coloring problem, we are given a graph and the question is to determine the smallest set of colors with which we can properly color the input graph. This problem is among Karp's famous list of 21 NP-complete problems [13] and since it often arises in practice, heuristics to solve it are deployed in a wide range of applications. A very natural such heuristic is the following. We greedily find a proper coloring of the graph, and then try to suppress any of its colors in the following way: say we want to suppress color c. If there is a vertex v that has received color c, and there is another color c = c that does not appear in the neighborhood of v, then we can safely recolor the vertex v with color c without making the coloring improper. We terminate this process once we cannot suppress any color anymore.
To predict the worst-case behavior of the above heuristic, Irving and Manlove defined the notions of a b-coloring and the b-chromatic number of a graph [12]. A b-coloring of a graph G is a proper coloring such that in every color class there is a vertex that has a neighbor in all of the remaining p k = ∆(G) + 1 − p k = m(G) − p 0 NP-c [14] NP-c [10] FPT [16,15] FPT [Thm. 4.3] O(k 5 )-vertex kernel [16,15]  and we give XP algorithms for the cases k = ∆(G) and k = m(G) − 1. We summarize these results in Table 1. Additionally, we prove that b-Coloring is FPT parameterized by k + k (G), where k (G) denotes the number of vertices of degree at least k. From the result of Kratochvíl et al. [14], it follows that b-Coloring is NP-complete even for the case when k (G) = 0. This is in contrast with the complexity of the Graph Coloring problem, in the sense that Aboulker et al. recently showed that the problem of deciding whether a graph admits a proper coloring with k colors is FPT parameterized by the number of vertices of degree at least k + 1, even when k is unbounded [1].

Preliminaries
We use the following notation: For k ∈ N, [k] . . = {1, . . . , k}. For a function f : X → Y and X ⊆ X, we denote by f | X the restriction of f to X and by f (X ) the set {f (x) | x ∈ X }. For a set X and an integer n, we denote by X n the set of all size-n subsets of X. Graphs. Throughout the paper a graph G with vertex set V (G) and edge set E(G) ⊆ V (G) 2 is finite and simple. We often denote an edge {u, v} ∈ E(G) by the shorthand uv. For graphs G and H we denote by H ⊆ G that H is a subgraph of G, i.e. V (H) ⊆ V (G) and E(H) ⊆ E(G). We often use the notation n . . = |V | and m . . = |E|. For a vertex v ∈ V (G), we denote by . When G is clear from the context, we abbreviate 'N G ' to 'N '. The degree of a vertex v ∈ V (G) is the size of its open neighborhood, and we denote it by deg G (v) . . = |N G (v)| or simply by deg(v) if G is clear from the context. For an integer k, we denote by k (G) the number of vertices of degree at least k in G.
For a vertex set be the subgraph of G obtained from removing the vertices in X and for a single vertex x ∈ V (G), we use the shorthand ' A graph G is said to be connected if for any 2-partition (X, Y ) of V (G), there is an edge xy ∈ E(G) such that x ∈ X and y ∈ Y , and disconnected otherwise. A connected component of a graph G is a maximal connected subgraph of G.
A graph G is a complete graph if every pair of vertices of G is adjacent.
A graph G is a bipartite graph if its vertex set can be partitioned into two independent sets. A bipartite graph with bipartition (A, B) with |A| = a and |B| = b is a complete bipartite graph if it has a · b edges and is denoted by K a,b . A star is the graph K 1,b , with b ≥ 2, and we call center the unique vertex of degree b and leaves the vertices of degree one.
Colorings. Given a graph G, a map γ : V (G) → [k] is called a coloring of G with k colors. If for every pair of adjacent vertices, uv ∈ E(G), we have that γ(u) = γ(v), then the coloring γ is called proper. For i ∈ [k], we call the set of vertices u ∈ V (G) such that γ(u) = i the color class i. If for all i ∈ [k], there exists a vertex x i ∈ V (G) such that γ(x i ) = i, and for each j ∈ [k], j = i, there is a neighbor y j ∈ N G (x i ) of x i such that γ(y j ) = j, then γ is called a b-coloring of G. In this case, we call the vertex x i the b-vertex of color i.
Parameterized Complexity. Let Σ be an alphabet. A parameterized problem is a set Π ⊆ Σ * ×N. A parameterized problem Π is said to be fixed-parameter tractable, or contained in the complexity class FPT, if there exists an algorithm that for each (x, k) ∈ Σ * × N decides whether (x, k) ∈ Π in time f (k)·|x| c for some computable function f and fixed integer c ∈ N. A parameterized problem Π is said to be contained in the complexity class XP if there is an algorithm that for all (x, k) ∈ Σ * ×N decides whether (x, k) ∈ Π in time f (k) · n g(k) for some computable functions f and g.
A kernelization algorithm for a parameterized problem Π ⊆ Σ * × N is a polynomial-time algorithm that takes as input an instance (x, k) ∈ Σ * ×N and either correctly decides whether (x, k) ∈ Π or outputs an instance (x , k ) ∈ Σ * × N with |x | + k ≤ f (k) for some computable function f for which (x, k) ∈ Π if and only if (x , k ) ∈ Π. We say that Π admits a kernel if there is a kernelization algorithm for Π.

Hardness Results
In this section we prove the hardness results of our complexity dichotomy. First, we show that b-Chromatic Number and b-Coloring are NP-complete for k = m(G) − 1 = ∆(G), based on a reduction due to Havet et al. [10] who showed NP-completeness for the case k = m(G). Proof. As in the proof of Havet et al. [10], the reduction is from the NP-complete problem 3-edge colorability of 3-regular graphs, which takes as input a 3-regular graph G and asks whether the edges of G can be properly colored with three colors.
Given an instance G of 3-edge colorability, an instance H of b-Chromatic Number and b-Coloring is constructed as follows. The graph H has one vertex for each vertex of G, that we denote by v 1 , . . . , v n , one vertex for each edge, that we denote by u 1 , . . . , u m and a set of 4n + 13 vertices that we denote by S. The edge set of H is such that H[{v 1 , . . . , v n }] is a clique, H[S] is the disjoint union of one copy of the complete bipartite graph K n,n+3 and two copies of K 2,n+3 and v i u j is an edge if the edge corresponding to u j is incident to the vertex corresponding to v i in G. The constructed graph H is such that ∆(H) = n + 3 and H has n + 4 vertices of degree n + 3, which implies that m(H) = n + 4. The difference to the construction used in [10] is that instead of the three complete bipartite graphs mentioned above, the authors use three copies of the star K 1,n+2 . Proof. Consider a component of H that induces a K i,n+3 , with i ∈ {2, n}. In any b-coloring with k ≥ n + 3 colors, only the vertices of degree n + 3 can be b-vertices in H. If x is a b-vertex for a given color, then the remaining k − 1 colors appear on the vertices of N (x). We conclude that any other vertex of degree n + 3 of this component will be assigned the same color as x.
We prove that H has a b-coloring with k = n+3 colors if and only if G is a Yes-instance for 3-edge colorability by using the same steps as in the proof of Theorem 3 of [10] and with the additional use of Claim 3.1.1. This proves the NP-completeness of b-Coloring when k = m(G) − 1 = ∆(G). Furthermore, we prove that χ b (H) ≥ n + 3 if and only if H has a b-coloring with n + 3 colors. This yields the analogous result for b-Chromatic Number.
First, assume that G is a Yes-instance for 3-edge colorability. Let E(G) = {e 1 , . . . , e m } and let γ E be a 3-edge coloring for G. We construct a b-coloring γ H for H in the following way.
Note that since γ E is a 3-edge coloring for G, the vertices v 1 , . . . , v n in H are b-vertices for the colors 4, . . . , n + 3. Now we can color the rest of the graph H in such a way that each connected component that is a complete bipartite graph contains a b-vertex for one of the three remaining colors. Now we consider the other direction. We start by observing that Claim 3.1.1 implies that H does not admit a b-coloring with n + 4 = m(H) = ∆(H) + 1 colors, since the set of vertices of degree n + 3 can contain b-vertices for at most three colors in any such a b-coloring. This implies that Assume H has a b-coloring γ H with n+3 colors. Since by Claim 3.1.1 the set S contains b-vertices for at most three colors, we have that the vertices v 1 , . . . , v n are b-vertices in this coloring. Moreover, since they induce a clique in H, they all have distinct colors. Assume, without loss of generality, that . . , u m }| = 3 for every i, we conclude that a 3-edge coloring for G can be obtained by assigning to each edge e i the color γ H (u i ). Now we show that b-Coloring remains NP-complete for fixed k if k = ∆(G) + 1 − p or k = m(G) − p for any p ≥ 2. To do so, we prove the following statement for k = 3, based on a reduction due to Sampaio [16].
Proof. Sampaio showed that the problem of deciding whether a graph G has a b-coloring with k colors is NP-complete for any fixed k ∈ N [16, Proposition 4.5.1]. For the case of k = 3, the reduction is from 3-coloring on planar 4-regular graphs which is known to be NP-complete [9]. In the proof of this proposition, the author adds three stars with 2 leaves each to the graph which can serve as the b-vertices in the resulting instance of b-Coloring. Since this does not increase the maximum degree, we immediately have that the problem of deciding whether a graph of maximum degree 4 has a b-coloring with 3 colors is NP-complete. Furthermore, by adding more leaves to one of the stars and thereby increasing the maximum degree of the graph in the resulting instance, we have that for any fixed integer c ≥ 1, it is NP-complete to decide whether a graph of maximum degree ∆(G) = c + 3 has a b-coloring with 3 colors.
Towards the statement regarding m(G), we first observe that for a 4-regular graph G on at least five vertices, we have that m(G) = 5. We observe that in any star with at least 2 leaves, the center vertex can be a b-vertex in a coloring with 3 colors. We construct a graph G by adding 5 stars with 4 leaves each to G, and we again have that G has a 3-coloring if and only if G has a b-coloring with 3 colors, showing that the problem of deciding wether a graph H with m(H) = 5 has a b-coloring with 3 colors, is NP-complete. Note that in this reduction, the center vertices of the stars can be regarded as the vertices determining the m-degree of the graph in the resulting instance of b-coloring with 3 colors, so we can extend this result in a similar way as above. That is, for any c ≥ 1, given a 4-regular graph G, we can add c + 4 stars with c + 4 − 1 leaves each to G, implying that for the resulting graph G , m(G ) = c + 4. Again, G has a 3-coloring if and only if G has a b-coloring with 3 colors, implying the second statement of the proposition.
We conclude this section by considering the complexity of the two problems on graphs with few vertices of high degree. Since b-Chromatic Number and b-Coloring are known to be NPcomplete when k = ∆ + 1 [14], we make the following observation which is of relevance to us since in Section 4.3, we show that b-Coloring is FPT parameterized by k + k (G).

Algorithms
In this section we present the algorithmic results of the paper. In Sections 4.1 and 4.2, we provide the algorithms for the complexity dichotomy and in Section 4.3 we give the FPT-algorithm for b-Coloring parameterized by the number of colors plus the number of high degree vertices.

FPT Algorithm for k = m(G)
The algorithm presented in this section makes use of the following reduction rule, which has already been applied in [16,15] to obtain the FPT algorithm for k = ∆(G) + 1.
Proof. First, we apply Reduction Rule 4.1 exhaustively to G. We consider the following 3-partition (D, T, R) of V (G).
-The vertices in D, called dense vertices, have degree at least k.
-The vertices in T , called tight vertices, have degree precisely k − 1.
Since we applied Reduction Rule 4.1 exhaustively, we make Observation 4.3.1. Every vertex in R has at least one neighbor in D ∪ T .
We pick an inclusion-wise maximal set B ⊆ D ∪ T such that for each pair of distinct vertices Case 1 (|B ∩ T | < k). 1 We show that for any vertex in u ∈ (T ∪ R) \ B, there is a vertex v ∈ B such that d(u, v) ≤ 4. Suppose u ∈ T . Since we did not include u in B, it immediately follows that there is some v ∈ B such that d(u, v) < 4. Now suppose u ∈ R. By Observation 4.3.1, u has a neighbor w in D ∪ T and by the previous argument, there is a vertex v ∈ B such that d(w, v) < 4. We conclude that d(u, v) ≤ 4.
Since the maximum degree of any vertex in , so we can solve the instance in time 2 O(k 5 ) · k O(1) using the algorithm of [15].
We show that we can construct a b-coloring γ : , we let γ(x i ) . . = i. Next, we color the vertices in D. Recall that |D| ≤ k, so we can color the vertices in D injectively with colors from [k], ensuring that this will not create a conflict on any edge in G In particular, there is no vertex in D that has two or more neighbors in B . To summarize, we can conclude that we can let γ color the vertices of D in such a way that: After this process, x i is a b-vertex for color i. We proceed in this way for all i ∈ [k]. Since for i, j ∈ [k] with i = j we have that d(x i , x j ) ≥ 4, it follows that there are no edges between N [x i ] and N [x j ] in G. Hence, we did not introduce any coloring conflict in the previous step. Now, all vertices in G that have not yet received a color by γ have degree at most k − 1, so we can extend γ to a proper coloring of G in a greedy fashion.
We summarize the whole procedure in Algorithm 1. We now analyze its runtime. Clearly, exhaustively applying Reduction condition of k = m(G) to k (G) ≤ k, we observe that the assumption k (G) ≤ k still guarantees that |D| ≤ k. Hence, in line 5, the bound of O(k 5 ) on V (G) remains the same and in line 11 we can find a coloring that is injective on D as well.  In this section we give XP-algorithms for the problems of deciding whether a given graph has a b-coloring with k = ∆ or k = m(G) − 1 colors, parameterized by k. To obtain these algorithms, we use a polynomial-time algorithm for a special case of the Precoloring Extension problem, formally defined as follows. A precoloring with k colors of a graph G is an assignment of colors to a subset of its vertices, i.e. for X ⊆ V (G), it is a map γ X : X → [k]. We say that a coloring γ : V (G) → [k] extends γ X , if γ| X = γ X .

Input:
A graph G, an integer k, and a precoloring γ X : X → [k] of a set X ⊆ V (G) Question: Does G have a proper coloring with k colors extending γ X ?
We show that we can use this theorem to give a polynomial-time algorithm for a slightly more general problem, namely when all high-degree vertices of the input graph are precolored.
Lemma 4.7. There is an algorithm that solves an instance (G, k, γ X ) of Precoloring Extension in polynomial time whenever max v∈V (G)\X deg(v) ≤ k.
Proof. First, we check whether γ X is a proper coloring of G[X] and if not, the answer is No. We create a new instance of Precoloring Extension (G , k, δ X ) as follows. For every vertex x ∈ X and every vertex y ∈ N G (x) \ X, we let x y be a new vertex that is only adjacent to y. We denote the set of these newly introduced vertices by X .
It is clear that (G, k, γ X ) is a Yes-instance of Precoloring Extension if and only if (G , k, δ X ) is a Yes-instance of Precoloring Extension. Furthermore, for every vertex z ∈ X , deg G (z) = 1 and for every vertex This means that we can solve the instance (G , k, γ X ) in polynomial time using Theorem 4.6. It is immediate that this algorithm is correct. We upper bound its runtime as follows. There are at most n k choices for B and for each x ∈ B, there are at most n k−1 choices of k − 1 neighbors of x. For each such set of neighbors we have to consider all bijective colorings of (k − 1) vertices with (k − 1) colors (to make x a b-vertex for its color) which amounts to (k − 1)!. Then, we have to guess at most k k+1 colorings for D, and apply the algorithm for Precoloring Extension of Lemma 4.7 which runs in time n O(1) . The total runtime is hence: as claimed.
We notice that we can use a simplified the algorithm provided in the proof of Theorem 4.8 to solve the question of whether a graph G has a b-coloring with k = ∆(G) colors as well. Following its notation, we have that D = ∅, since there is no vertex of degree at least k + 1 in G. The remainder of the algorithm remains the same.
However, we can observe an improvement of the upper bound on the runtime of the resulting algorithm: When we guess the set of b-vertices B, we have that each vertex x ∈ B has degree either k or k − 1. Hence we have at most k k−1 = k choices for size-(k − 1) sets of neighbors of x that can be used to make x a b-vertex. The runtime of the algorithm becomes: (1) Theorem 4.9. There is an algorithm that decides whether a graph G has a b-coloring with k = ∆(G) colors in time 2 O(k 2 log k) · n k+O(1) .

FPT Algorithm for k +
Since b-Coloring is NP-complete for every fixed k and, by Observation 3.3, b-Coloring is also NP-complete on graphs with k (G) = 0, we consider a parameterization by k + k (G). We show that in this case, the problem is in FPT. Note that the algorithm we provide in this section can be used to solve the case of k = m(G) for which we gave a separate algorithm in Section 4.1, see Algorithm 1. However, Algorithm 1 is much simpler than the algorithm presented in this section, and simply applying the following algorithm for the case k = m(G) results in a runtime of Proof. The overall strategy of the algorithm is similar to Algorithm 1. First we note that if ≤ k, then we can apply Algorithm 1 directly to solve the instance at hand, see Remark 4.4. Hence we can assume throughout the following that > k. Again, we consider a partition (D, T, R) of V (G), where the vertices in D have degree at least k, the vertices in T have degree k − 1 and the vertices in R have degree less than k − 1. We assume that Reduction Rule 4.1 has been applied exhaustively, so Observation 4.3.1 holds, i.e. every vertex in R has at least one neighbor in D ∪ T . Now, we pick an inclusion-wise maximal set B ⊆ D ∪ T such that for each pair of distinct Case 1 (|B ∩ T | < k). By the same argument given in Case 1 of the proof of Theorem 4.3, we have that any vertex in T ∪ R is at distance at most 2 from a vertex in B and we can conclude that in this case, |V (G)| = k O( ) + = k O( ) , so we can solve the instance in time 2 k O( ) · k O( ) by [15]. , and then modify it so that can be extended to a b-coloring of G. In this process we will be able to guarantee for each i ∈ [k], that either x i can be the b-vertex for color i, or we will have found another vertex in D that can serve as the b-vertex of color i. The difficulty here arises from the following situation: Suppose that in the coloring we computed for G[D], a vertex x i has two neighbors in D that received the same color. Then, x i cannot be the b-vertex of color i in any extension of that coloring, since  Proof. Suppose that for some i ∈ [k], x i ∈ V (C), and for the sake of a contradiction that for some j = i, x j ∈ V (C). It follows that there are vertices y i ∈ N (x i ) ∩ D and y j ∈ N (x j ) ∩ D, such that there is a path in G[D ∪ B ] − {x i , x j } from x i to y j . However, since |D ∪ B | = + k, this path has length at most + k − 3, implying that d(x i , x j ) ≤ + k − 1 < 2 , a contradiction with the choice of B.
Let furthermore C ∅ be the set of connected components of G[D ∪ B ] that do not contain any vertex from B . We observe that any proper coloring of G[D ∪ B ] can be obtained from independently coloring the vertices in C 1 , . . . , C k , and C ∅ . If for some i ∈ [k], C i is a trivial 2 component, then N (x i ) ∩ D = ∅. Hence, we can assign x i any color without creating any conflict with the remaining vertices in G[D ∪ B ]. We illustrate the structure of G in Figure 1. Before we proceed with the proof of the next claim, we introduce some notation. For X ⊆ V (G), a coloring γ : X → [k], and i, j ∈ [k], we denote by γ i↔j the coloring obtained from γ by switching colors i and j, i.e. for v ∈ X we let: Proof. We can assume that γ(x i ) = i, otherwise we let γ . . = γ i↔j for some [k] j = i. If γ is injective on N C i [x i ], we let δ . . = γ and we are in case (ii).
Otherwise, we do as follows. Let j ∈ [k] with j = i be a color that does not appear on any vertex in N C i (x i ), i.e. there is no vertex y ∈ N C i (x i ) such that γ(y) = j. Such a color must exists by the fact that γ is not injective on N C i [x i ] and the fact that deg G (x i ) = k − 1. For each vertex z ∈ V (C i ) with γ(z) = j, we do the following. for color j. We let δ . . = γ i↔j and we are in case (i).
2) Otherwise, there is a color j = j that does not appear in the neighborhood of z. We update γ by setting γ(z) . . = j , keeping the coloring γ proper.
If these two steps are executed for all vertices that γ colored j without ending up in case (i), then γ is a proper coloring of C i with colors [k] \ {j}. Now, let y 1 , y 2 ∈ N C i (x i ) be a pair of non-adjacent neighbors of x i . We add the edge y 1 y 2 to C i and update γ(y 1 ) . . = j. Now, γ is a proper k-coloring of the graph obtained from C i by adding an edge in the neighborhood of x i . We repeat this process until we either reached case (i) at some stage, or we have that γ is a proper k-coloring of the graph obtained from C i by making N C i [x i ] a clique. The latter case implies that γ is injective on N C i [x i ] and we are in case (ii) by letting δ . . = γ. This recoloring procedure terminates within time O( k−1 The algorithm to solve this case now works as follows. First, we compute a proper k-coloring γ of G[D ∪ B ]. We derive from γ another k-coloring δ of G[D ∪ B ]. For each i ∈ [k], we do the following. If C i is a nontrivial component, then, with input γ| V (C i ) we compute a proper k-coloring δ i of C i using Claim 4.10.2 satisfying the stated conditions, and let δ| V (C i ) . . = δ i . Finally, we let δ| V (C ∅ ) . . = γ| V (C ∅ ) . We now show how to extend the coloring δ to a b-coloring of the entire graph G. Let i ∈ [k]. By Claim 4.10.2 we know that in δ either there is a b-vertex for color i in C i or δ is injective on In the latter case, let N G (x i ) = {y 1 , . . . , y k−1 } and assume wlog. that for some k ≤ k − 1, N C i (x i ) = {y 1 , . . . , y k }. Then, k different colors appear on {y 1 , . . . , y k } and we can let δ color {y k +1 , . . . , y k−1 } bijectively with the remaning k − 1 − k colors to make x i the b-vertex of color i. (Note that since d(x i , x j ) ≥ 2 for i = j, this does not introduce any coloring conflict.) The remaning vertices of G have degree at most k − 1, so we can extend the coloring δ greedily to the remainder of G.
It remains to argue the runtime of the algorithm. Applying Reduction Rule 4.1 exhaustively can be done in time n O(1) . As mentioned above, in Case 1 we can solve the instance in time 2 k O( ) · k O( ) . In Case 2, we can compute a proper k-coloring of G[D ∪ B ] in time O * (2 +k ) = 2 O( ) using standard methods [3]. Modifying this coloring to satisfy the condition of Claim 4.10.2 for Extending the coloring to the remainder of G can be done in time n O(1) , so the total runtime of the algorithm is 2 k O( ) · k O( ) + 2 O( ) + O( 5 ) + n O(1) = 2 k O( ) · k O( ) + n O(1) , as claimed.
Similar to above, we have the following consequence.
Corollary 4.11. The problem of deciding whether a graph G admits a b-coloring with k colors admits a kernel on k O( ) vertices, where . . = k (G).

Conclusion
We have presented a complexity dichotomy for b-Coloring with respect to two upper bounds on the b-chromatic number, in the following sense: We have shown that given a graph G and for fixed k ∈ {∆(G) + 1 − p, m(G) − p}, it can be decided in polynomial time whether G has a b-coloring with k colors whenever p ∈ {0, 1} and the problem remains NP-complete whenever p ≥ 2, already for k = 3. Furthermore, we gave an FPT-algorithm for b-Coloring parameterized by k + k (G), where k (G) denotes the number of vertices of degree at least k in G.
The most immediate question left open in this work is the parameterized complexity of the cases when k = ∆(G) or k = m(G) − 1. In both of them, we have provided XP-algorithms, and it would be interesting to see whether these problems are FPT or W-hard.
Recently, Effantin et al. [8] introduced the relaxed b-chromatic number of a graph G, χ r b (G), as the maximum b-chomatic number of any induced subgraph of G, i.e. χ r b (G) . . = max X⊆V (G) χ b (G[X]). It is clear that χ b (G) ≤ χ r b (G), so it would be interesting to see if for fixed k, the problem of deciding whether a graph G admits a b-coloring with k colors when the value of k is close to χ r b (G) admits a similar dichotomy as the ones we presented for the upper bounds ∆(G) + 1 and m(G) on χ b (G).