Additive non-approximability of chromatic number in proper minor-closed classes

Robin Thomas asked whether for every proper minor-closed class C, there exists a polynomial-time algorithm approximating the chromatic number of graphs from C up to a constant additive error independent on the class C. We show this is not the case: unless P=NP, for every integer k>=1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k>=1 and 1<=\beta<=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta.chi(G)+(4-3beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

The problem of determining the chromatic number, or even of just deciding whether a graph is colorable using a fixed number c ≥ 3 of colors, is NP-complete [7], and thus it cannot be solved in polynomial time unless P = NP. Even the approximation version of the problem is hard: for every ε > 0, Zuckerman [18] proved that unless P = NP, there exists no polynomial-time algorithm approximating the chromatic number of an nvertex graph within multiplicative factor n 1−ε .
There are more restricted settings in which the graph coloring problem becomes more tractable. For example, the well-known Four Color Theorem implies that deciding c-colorability of a planar graph is trivial for any c ≥ 4; still, 3-colorability of planar graphs is NP-complete [7]. From the approximation perspective, this implies that chromatic number of planar graphs can be approximated in polynomial time up to multiplicative factor of 4/3 (but not better), and additively up to 1. More generally, the result of Thomassen [16] on 6-critical graphs implies that the c-coloring problem restricted to graphs that can be drawn in any fixed surface of positive genus is polynomial-time solvable for any c ≥ 5. The case c = 3 includes 3-colorability of planar graphs and consequently is NP-complete, while the complexity of 4-colorability of embedded graphs is unknown for all surfaces of positive genus. Consequently, chromatic number of graphs embedded in a fixed surface can be approximated up to multiplicative factor of 5/3 and additively up to 2. If a graph can be drawn in a given surface, all its minors can be drawn there as well. Hence, it is natural to also consider the coloring problem in the more general setting of proper minor-closed classes. Further motivation for this setting comes from Hadwiger's conjecture, stating that all K k -minor-free graphs are (k − 1)-colorable. This conjecture is open for all k ≥ 7, and not even a polynomial-time algorithm to decide (k − 1)-colorability of K k -minorfree graphs is known (Kawarabayashi and Reed [9] designed an algorithm that for a given input graph G finds a (k − 1)-coloring, or a minor of K k in G, or finds a counterexample to Hadwiger's conjecture). However, any K kminor-free graph is O(k √ log k)-colorable [11]. This implies that for every proper minor-closed class G, if k is the minimum integer such that K k ∈ G, then there exists a constant c = O(k √ log k) such that every graph in G is c-colorable, and thus chromatic number of graphs in G can be approximated up to multiplicative factor c/3 and additively up to c − 3.
On the hardness side, consider for any planar graph G and an integer t ≥ 0 the graph G t obtained from G by adding t universal vertices (adjacent to every other vertex of G t ). Then χ(G t ) = χ(G) + t, and since 3-colorability of planar graphs is NP-complete, there cannot exist a polynomial-time algorithm to decide whether such a graph G t is (t + 3)-colorable, unless P = NP. Furthermore, G t does not contain K t+5 as a minor; indeed, each minor of G t has an induced planar subgraph containing all but t of its vertices, which is not the case for K t+5 . This outlines the importance of another graph parameter in this context, the apex number : we say a graph H is t-apex if there exists a set X of vertices of H of size at most t such that H − X is planar, and the apex number of H is the minimum t such that H is t-apex. The presented construction shows that if the apex number of H is at least t, then (t + 2)-colorability is NP-complete even when restricted to the class of Hminor-free graphs. On the positive side, Dvořák and Thomas [5] gave, for any t-apex graph H and integer c ≥ t + 4, a polynomial-time algorithm to decide whether a (t + 3)-connected H-minor-free graph is c-colorable (the connectivity assumption is necessary, since they also proved that for every integer t ≥ 1, there exists a t-apex graph H such that testing (t + 4)-colorability of (t + 2)-connected H-minor-free graphs is NP-complete).
In all the mentioned results for proper minor-closed classes, the number of colors needed and thus also the magnitude of error of the corresponding approximation algorithms depended on the specific class. This contrasts with the case of embedded graphs: the multiplicative (5/3) and additive (2) errors of these approximation algorithms are independent on the fixed surface in that the graphs are drawn. Hence, it is natural to ask the following questions. Question 1. Does there exist β ≥ 1 with the following property: for every proper minor-closed class G, there exists a polynomial-time algorithm taking as an input a graph G ∈ G and returning an integer c such that χ(G) ≤ c ≤ βχ(G)? Question 2. Does there exist α ≥ 0 with the following property: for every proper minor-closed class G, there exists a polynomial-time algorithm taking as an input a graph G ∈ G and returning an integer c such that χ(G) ≤ c ≤ χ(G) + α? That is, is it possible to approximate chromatic number up to a multiplicative or additive error independent on the considered class of graphs G, as long as G is proper minor-closed? Perhaps a bit surprisingly, the answer to Question 1 is positive. As shown by DeVos et al. [4] and algorithmically by Demaine et al. [3], for every proper minor-closed class G, there exists a constant γ G such that the vertex set of any graph G ∈ G can be partitioned in polynomial time into two parts A and B with both G[A] and G[B] having tree-width at most γ G . Consequently, χ(G[A]), χ(G[B]) ≤ χ(G) can be determined exactly in linear time [1], and we can color G[A] and G[B] using disjoint sets of colors, obtaining a coloring of G using at most 2χ(G) colors. That is, β = 2 has the property described in Question 1. In the light of this result, Question 2 may seem more tractable. Thomas [15] conjectured that such a constant α exists, and Kawarabayashi et al. [10] conjectured that this is the case even for list coloring. As our main result, we disprove these conjectures.
Theorem 3. Let k 0 be a positive integer, let F be a (4k 0 − 3)-connected graph that is not (4k 0 − 4)-apex, and let 1 ≤ β ≤ 4/3 be a real number. Unless P = NP, there is no polynomial-time algorithm taking as an input an F -minor-free graph G and returning an integer c such that χ(G) ≤ c < βχ(G) + (4 − 3β)k 0 .
In particular, in the special case of β = 1 and F being a clique, we obtain the following.
Corollary 4. Let k 0 be a positive integer. Unless P = NP, there is no polynomial-time algorithm taking as an input a K 4k 0 +1 -minor-free graph G and returning an integer c such that On the positive side, Kawarabayashi et al. [10] showed it is possible to approximate chromatic number of K k -minor free graphs in polynomial time additively up to k − 2. We leave open the question whether a better additive approximation (of course above the bound ≈ k/4 given by Corollary 4) is possible.
Another positive result was given by Demaine et al. [2], who proved that if H is a 1-apex graph, then the chromatic number of H-minor-free graphs can be approximated additively up to 2. Let us also remark that if H is 0-apex (i.e., planar), then H-minor-free graphs have bounded tree-width [13], and thus their chromatic number can be determined exactly in linear time [1]. We generalize these results to excluded minors with larger apex number (the relevance of the apex number in the context is already showcased by Theorem 3).
Theorem 5. Let t be a positive integer and let H be a t-apex graph. There exists a polynomial-time algorithm taking as an input an H-minor-free graph G and returning an integer c such that χ(G) ≤ c ≤ χ(G) + t + 3.
The construction we use to establish Theorem 3 results in graphs with large clique number (on the order of k 0 ). On the other hand, forbidding triangles makes the coloring problem for embedded graphs more tractableall planar graphs are 3-colorable [8] and there exists a linear-time algorithm to decide 3-colorability of a graph embedded in any fixed surface [6]. It is natural to ask whether Question 2 could not have a positive answer for triangle-free graphs, and this question is still open. On the negative side, we show that forbidding cliques of size 4 is not sufficient.
Theorem 6. Let β and d be real numbers such that 1 ≤ β < 4/3 and d ≥ 0. Let m = ⌈d/(4 − 3β)⌉. There exists a positive integer k 0 = O m 4 log 2 m such that the following holds. Let F be a (4k 0 − 1)-connected graph with at least 4k 0 + 8 vertices that is not (4k 0 − 4)-apex. Unless P = NP, there is no polynomial-time algorithm taking as an input an F -minor-free graph G with ω(G) ≤ 3 and returning an integer c such that χ(G) ≤ c < βχ(G) + d.
In particular, in the special case of β = 1 and F being a complete graph, we get the following. 0 / log 1/2 k 0 ) as follows. Unless P = NP, there is no polynomial-time algorithm taking as an input a K 4k 0 +8 -minor-free graph G with ω(G) ≤ 3 and returning an integer c such that χ(G) ≤ c ≤ χ(G) + d.
On the positive side, we offer the following small improvement to the additive error of Theorem 5.
Theorem 8. Let t be a positive integer and let H be a t-apex graph. There exists a polynomial-time algorithm taking as an input an H-minor-free graph G with no triangles and returning an integer c such that χ(G) ≤ c ≤ χ(G) + ⌈(13t + 172)/14⌉.
What about graphs of larger girth? It turns out that Question 2 has positive answer for graphs of girth at least 5, with α = 6. Somewhat surprisingly, it is not even necessary to forbid triangles to obtain this result, just forbidden 4-cycles are sufficient. Indeed, we can show the following stronger result.
Theorem 9. Let a ≤ b be positive integers and let G be a proper minor-closed class of graphs. There exists a polynomial-time algorithm taking as an input a graph G ∈ G not containing K a,b as a subgraph and returning an integer c such that χ(G) ≤ c ≤ χ(G) + a + 4.
Let us remark that the multiplicative 2-approximation algorithm of Demaine et al. [3] can be combined with the algorithms of Theorems 5, 8, and 9 by returning the minimum of their results. E.g., if H is a t-apex graph, then there is a polynomial-time algorithm coloring an H-minor-free graph G using at most min(2χ(G), χ(G) + t + 3) ≤ βχ(G) + (2 − β)(t + 3) colors, for any β such that 1 ≤ β ≤ 2; the combined multiplicative-additive non-approximability bounds of Theorems 3 and 6 are also of interest in this context.
In Section 1, we present a graph construction which we exploit to obtain the non-approximability results in Section 2. The approximation algorithms are presented in Section 3.

Tree-like product of graphs
Let G and H be graphs, and let |V (G)| = n and V (H) = {u 1 , . . . , u k }. Let T n,k denote the rooted tree of depth k + 1 such that each vertex at depth at most k has precisely n children (the depth of the tree is the number of vertices of a longest path starting with its root, and the depth of a vertex x is the number of vertices of the path from the root to x; i.e., the root has depth 1). For each non-leaf vertex x ∈ V (T n,k ), let G x be a distinct copy of the graph G and let θ x be a bijection from V (G x ) to the children of x in T n,k .
If v ∈ V (G x ), y is a non-leaf vertex of the subtree of T n,k rooted in θ x (v), and z ∈ V (G y ), then we say that v is a progenitor of z. The level of v is defined to be the depth of x in T n,k . Note that a vertex at level j has exactly one progenitor at level i for all positive i < j. The graph T (G, H) is obtained from the disjoint union of the graphs G x for non-leaf vertices x ∈ V (T n,k ) by, for each edge u i u j ∈ E(H) with i < j, adding all edges from vertices of T (G, H) at level j to their progenitors at level i. Note that the graph T (G, H) depends on the ordering of the vertices of H, which we consider to be fixed arbitrarily.
Consider a clique K in T (G, H), and let v be a vertex of K of largest level. Let x be the vertex of T n,k such that v ∈ V (G x ). Note that all vertices of K \ V (G x ) are progenitors of v, and the vertices of H corresponding to their levels are pairwise adjacent. Consequently, For i = 0, . . . , k − q, let G i denote the graph obtained from G by adding i universal vertices. Observe that T (G, H) is obtained from copies of G 0 , . . . , G k−q by clique-sums on cliques of size at most k − q (consisting of the progenitors whose level is most k − q). Hence, each (k − q + 1)-connected minor F of T (G, H) is a minor of one of G 0 , . . . , G k−q , and thus a minor of G can be obtained from F by removing at most k − q vertices.
For an integer p ≥ 1, the p-blowup of a graph H 0 is the graph H obtained from H 0 by replacing every vertex u by an independent set S u of p vertices, and by adding all edges zz ′ such that z ∈ S u and z ′ ∈ S u ′ for some uu ′ ∈ E(H 0 ). For the purposes of constructing the graph T (G, H), we order the vertices of H so that for each u ∈ V (H 0 ), the vertices of S u are consecutive in the ordering. The strong p-blowup is obtained from the p-blowup by making the sets S u into cliques for each u ∈ V (H 0 ). For integers a ≥ b ≥ 1, an (a : b)-coloring of H 0 is a function ϕ that to each vertex of H 0 assigns a subset of {1, . . . , a} of size b such that ϕ(u) ∩ ϕ(v) = ∅ for each edge uv of H 0 . The fractional chromatic number χ f (H 0 ) is the infimum of {a/b : H 0 has an (a : b)-coloring}. Note that if H is the strong p-blowup of a graph H 0 , then a c-coloring of H gives a (c : p)-coloring of H 0 . Consequently, we have the following.
We now state a key result concerning the chromatic number of the graph T (G, H).

Lemma 12.
Let p, c ≥ 1 be integers and let G be a graph. Let H 0 be a graph such that χ(H 0 ) = χ f (H 0 ) = c, and let H be the p-blowup of H 0 . Then Let ϕ H be a proper coloring of H using c colors. Let C 1 , . . . , C c be pairwise disjoint sets of χ(G) colors. For each non-leaf vertex x of T n,k of depth i, color G x properly using the colors in C ϕ H (u i ) . Observe that this gives a proper coloring of T (G, H) using at most cχ(G) colors, and thus Suppose now that χ(G) ≥ p and consider a proper coloring ϕ of T (G, H).
. . x k+1 be a path in T n,k from its root x 1 to one of the leaves and let ψ be a coloring of H constructed as follows. Suppose that we already selected Note that ψ is a proper coloring of H such that for each u ∈ V (H 0 ), ψ assigns vertices in S u pairwise distinct colors. Consequently, ψ is a proper coloring of the strong p-blowup of H 0 , and thus ψ (and ϕ) uses at least pχ f (H 0 ) = cp distinct colors by Observation 11. We conclude that χ(T (G, H)) ≥ cp.
For positive integers p and k, let K k×p denote the p-blowup of K k , i.e., the complete k-partite graph with parts of size p. Let us summarize the results of this section in the special case of the graph T (G 0 , K k×4 ) with G 0 planar. Corollary 13. Let G 0 be a planar graph with n vertices and let k 0 be a positive integer.
form an independent set. The claims follow from Lemma 10 (with H = K k 0 ×4 , k = 4k 0 and q = 4, using the fact that every minor of a planar graph is planar) and Lemma 12 (with H 0 = K k 0 , p = 4 and c = k 0 ).

Non-approximability
The main non-approximability result is a simple consequence of Corollary 13 and NP-hardness of testing 3-colorability of planar graphs.
Proof of Theorem 3. Suppose for a contradiction that there exists such a polynomial-time algorithm A, taking as an input an F -minor-free graph G and returning an integer c such that χ(G) ≤ c < βχ(G) + (4 − 3β)k 0 .
Let G 0 be a planar graph, and let G = T (G 0 , K k 0 ×4 ). By Corollary 13, the size of G is polynomial in the size of G 0 and G is F -minor-free. Furthermore, if G 0 is 3-colorable, then χ(G) ≤ 3k 0 , and otherwise χ(G) ≥ 4k 0 . Hence, if G 0 is 3-colorable, then the value returned by the algorithm A applied to G is less than βχ(G) + (4 − 3β)k 0 ≤ 4k 0 , and if G 0 is not 3-colorable, then the value returned is at least χ(G) ≥ 4k 0 . This gives a polynomial-time algorithm to decide whether G 0 is 3-colorable.
However, it is NP-hard to decide whether a planar graph is 3-colorable [7], which gives a contradiction unless P = NP.
Note that the graphs T (G 0 , K k 0 ×4 ) used in the proof of Theorem 3 have large cliques (of size greater than k 0 ). This turns out not to be essential-we can prove somewhat weaker non-approximability result even for graphs with clique number 3. To do so, we need to apply the construction with both G and H 0 being triangle-free. A minor issue is that testing 3-colorability of triangle-free planar graphs is trivial by Grötzsch' theorem [8]. However, this can be easily worked around.

Lemma 14.
Let G denote the class of graphs such that all their 3-connected minors with at least 12 vertices are planar. The problem of deciding whether a triangle-free graph G ∈ G is 3-colorable is NP-hard. Proof. Let R 0 be the Grötzsch graph (R 0 is a triangle-free graph with 11 vertices and chromatic number 4, and all its proper subgraphs are 3-colorable). Let R be a graph obtained from R 0 by removing any edge uv. Note that R is 3-colorable and the vertices u and v have the same color in every 3-coloring.
Let G 1 be a planar graph. Let G 2 be obtained from G 1 by replacing each edge xy of G 1 by a copy of R whose vertex u is identified with x and an edge is added between v and y (i.e., G 2 is obtained from G 1 by a sequence of Hajós sums with copies of R 0 ). Clearly, G 2 is triangle-free, it is 3-colorable if and only if G 1 is 3-colorable, and |V (G 2 )| = |V (G 1 )| + 10|E(G 1 )|. Furthermore, G 2 is obtained from a planar graph by clique-sums with R 0 on cliques of size two, and thus every 3-connected minor of G 2 is either planar or a minor of R 0 (and thus has at most 11 vertices). Hence, G 2 belongs to G.
Since testing 3-colorability of planar graphs is NP-hard, it follows that testing 3-colorability of triangle-free graphs from G is NP-hard.
We also need a graph H 0 which is triangle-free and its fractional chromatic number is large and equal to its ordinary chromatic number. We are not aware of such a construction being previously studied; in Appendix, we prove the following. Consider a triangle-free graph G 0 ∈ G, and let G = T (G 0 , H). By Lemma 10, the size of G is polynomial in the size of G 0 . Consider any (4k 0 − 1)-connected minor F ′ of G. By Lemma 10, there exists a set X of size at most 4k 0 − 4 such that F ′ − X is a minor of G 0 . Since F ′ − X is 3connected and G 0 ∈ G, we conclude that either |V (F ′ )| ≤ |X| + 11 ≤ 4k 0 + 7 or F ′ − X is planar. Consequently, F ′ = F , and thus G does not contain F as a minor. Furthermore, ω(G) ≤ ω(G 0 ) + ω(H) − 1 ≤ 3.
Recall that χ(H m ) = χ f (H m ) = m. By Lemma 12, if G 0 is 3-colorable, then χ(G) ≤ 3m, and otherwise χ(G) ≥ 4m. Hence, if G 0 is 3-colorable, then the value returned by the algorithm A applied to G is less than βχ(G) + d ≤ 3βm + d ≤ 4m, and if G 0 is not 3-colorable, then the value returned is at least χ(G) ≥ 4m. This gives a polynomial-time algorithm to decide whether G 0 is 3-colorable, in contradiction to Lemma 14 unless P = NP.

Approximation algorithms
Let us now turn our attention to the additive approximation algorithms. The algorithms we present use ideas similar to the ones of the 2-approximation algorithm of Demaine et al. [3] and of the additive approximation algorithms of Kawarabayashi et al. [10] and Demaine et al. [2]. We find a partition of the vertex set of the input graph G into parts L and C such that G[L] has bounded tree-width (and thus its chromatic number can be determined exactly) and G[C] has bounded chromatic number, and color the parts using disjoint sets of colors. The existence of such a decomposition is proved using the minor structure theorem [14], in the variant limiting the way apex vertices attach to the surface part of the decomposition. The proof of this stronger version can be found in [5]. Let us now introduce definitions necessary to state this variant of the structure theorem.
A tree decomposition of a graph G is a pair (T, β), where T is a tree and β is a function assigning to each vertex of T a subset of V (G), such that for each uv ∈ E(G) there exists z ∈ V (T ) with {u, v} ⊆ β(z), and such that for each v ∈ V (G), the set {z ∈ V (T ) : v ∈ β(z)} induces a non-empty connected subtree of T . The width of the tree decomposition is max{|β(z)| : z ∈ V (T )} − 1, and the tree-width of a graph is the minimum of the widths of its tree decompositions.
The decomposition is rooted if T is rooted. For a rooted tree decomposition (T, β) and a vertex z of T distinct from the root, if w is the parent of z in T , we write β ↑ z := β(z) ∩ β(w) and β ↓ z := β(z) \ β(w). If z is the root of T , then β ↑ z := ∅ and β ↓ z := β(z). The torso expansion of a graph G with respect to its rooted tree decomposition (T, β) is the graph obtained from G by adding edges of cliques on β ↑ z for all z ∈ V (T ).
A path decomposition is a tree decomposition (T, β) where T is a path. Theorem 16. For every non-negative integer t and a t-apex graph H, there exists a constant a H such that the following holds. For every H-minor-free graph G, there exists a rooted tree decomposition (T, β) of G with the following properties. Let G ′ denote the torso expansion of G with respect to (T, β).
the graphs G v,i and G v,j are vertex-disjoint and G ′ contains no edges between V (G v,i ) and V (G v,j ), Furthermore, the tree decomposition and the sets and subgraphs as described can be found in polynomial time.
Informally, the graph G is a clique-sum of the graphs G ′ [β(v)] for v ∈ V (T ), and β(v) contains a bounded-size set A v of apex vertices such that G ′ [β(v)] − A v can be embedded in Σ v up to a bounded number of vortices of bounded depth. Furthermore, at most t − 1 of the apex vertices (forming the set A ′ v ) can have neighbors in the part G v of G ′ [β(v)] − A v drawn in Σ v , or in the other bags of the decomposition that intersect G v . Note that it is also possible that Σ v is the null surface, and consequently A v = β(v).
We need the following observation on graphs embedded up to vortices.
Lemma 17. Let Σ be a surface of Euler genus g and let a be a non-negative integer. Let G be a graph and let G 0 , G 1 , . . . , G m be its subgraphs such that Proof. If Σ is the sphere, then we can set L 0 = ∅; hence, we can assume that g ≥ 1. Let G ′ 0 be the graph obtained from G 0 by, for 1 ≤ i ≤ m, adding a new vertex r i drawn inside f i and joined by edges to all vertices of V (G 0 ) ∩ V (G i ). Note that G ′ 0 has a 2-cell embedding in Σ extending the embedding of G. Let F be a subgraph of G ′ 0 such that the embedding of F in Σ inherited from the embedding of G ′ 0 is 2-cell and |V (F )| + |E(F )| is minimum. Then F has only one face, since otherwise it is possible to remove an edge separating distinct faces of F , and F has minimum degree at least two, since otherwise we can remove a vertex of degree at most 1 from F . By generalized Euler's formula, we have |E(F )| = |V (F )| + g − 1, and thus F contains at most 2(g − 1) vertices of degree greater than two. By considering the graph obtained from F by suppressing vertices of degree two, we see that F is either a cycle (if g = 1) or a subdivision of a graph with at most 3(g − 1) edges.
Let M 0 be the set of vertices of F of degree at least three and their neighbors. We claim that each vertex of V (F ) \ M 0 is adjacent in G ′ 0 to only two vertices of V (F ). Indeed, suppose that a vertex x ∈ V (F ) \ M 0 has at least three neighbors in G ′ 0 belonging to V (F ). Let w and y be the neighbors of x in F , and let z be a vertex distinct from w and y adjacent to x in G ′ 0 . The graph F + xz has two faces, and by symmetry, we can assume that the edge xy separates them. Since x ∈ M 0 , the vertex y has degree two in F , and thus the embedding of F + xz − y is 2-cell, contradicting the minimality of F .
Let M 1 be the set of vertices of F at distance at most 4 from a vertex of degree greater than two. Note that |M 1 | ≤ 26g. For 1 ≤ i ≤ m, let N i denote the set of vertices of F − M 1 that are in G ′ 0 adjacent to a vertex of V (G i ) ∩ V (G 0 ) \ V (F ). We claim that |N i | ≤ 9. Indeed, suppose for a contradiction that |N i | ≥ 10 and consider a path w 4 w 3 w 2 w 1 xy 1 y 2 y 3 y 4 in F such that x belongs to N i (vertices x, w 1 , . . . , w 4 , y 1 , . . . , y 4 have degree two in F , since x ∈ M 1 ). If r i ∈ V (F ), then let Q be a path in G ′ 0 of length two between x and r i through a vertex of V (G i ) ∩ V (G 0 ) \ V (F ); note that r i ∈ {w 1 , w 2 , y 1 , y 2 }, since r i has at least 10 neighbors in V (F ) and belongs to M 0 by the previous paragraph and x ∈ M 1 . If r i ∈ V (F ), then there exists a vertex z ∈ N i \ {x, w 1 , . . . , w 4 , y 1 , . . . , y 4 }; we let Q be a path of length at most four between x and z passing only through their neighbors in V (G i ) ∩ V (G 0 ) \ V (F ) and possibly through r i . By symmetry, we can assume that the edge xy 1 separates the two faces of F + Q, and the graph F +Q−{y 1 , y 2 } if r i ∈ V (F ) or F +Q−{y 1 , y 2 , y 3 , y 4 } if r i ∈ V (F ) contradicts the minimality of F .
Let G ′ be the graph obtained from G 1 ∪ . . . ∪ G m by adding vertices of M as universal ones, adjacent to all other vertices of G ′ . The tree-width of G ′ is at most a + |M| ≤ 26g + 9m + a. Note that each vertex of Note that the set L 0 can be found in polynomial time. For the clarity of presentation of the proof we selected F with |V (F )| + |E(F )| minimum; however, it is sufficient to start with an arbitrary inclusionwise-minimal subgraph with exactly one 2-cell face (obtained by repeatedly removing edges that separate distinct faces and vertices of degree at most 1) and repeatedly perform the reductions described in the proof until each vertex of V (F )\M 0 is adjacent in G ′ 0 to only two vertices of V (F ) and until |N i | ≤ 9 for 1 ≤ i ≤ m. For positive integers t and a, we say that a rooted tree decomposition (T, β) of a graph G is (t, a)-restricted if for each vertex v of T , the subgraph of the torso expansion of G induced by β ↓ v is planar, |β ↑ v| ≤ t − 1, and each vertex of β ↓ v has at most a − 1 neighbors in G that belong to β ↑ v. Using the decomposition from Theorem 16, we now partition the considered graph into a part of bounded tree-width and a (t, t)-restricted part.
Theorem 18. For every positive integer t and a t-apex graph H, there exists a constant c H with the following property. The vertex set of any H-minorfree graph G can be partitioned in polynomial time into two parts L and C such that G[L] has tree-width at most c H and G[C] has a (t, t)-restricted rooted tree decomposition. Additionally, for any such graph H and positive integers a ≤ b, there exists a constant c H,a,b such that if G is H-minor-free and does not contain K a,b as a subgraph, then L and C can be chosen so that G[L] has tree-width at most c H,a,b and G[C] has a (t, a)-restricted rooted tree decomposition. Proof. Let a H be the constant from Theorem 16 for H, and let g be the maximum Euler genus of a surface in that H cannot be embedded. Let c H = 26g + 11a H and c H,a,b = c H + t−1 a (b − 1). Since G is H-minor-free, we can in polynomial time find its rooted tree decomposition (T, β), its torso expansion G ′ , and for each v ∈ V (T ), find A v , . . , G v,m , and Σ v as described in Theorem 16. Let L ′ v be the set of vertices obtained by applying Lemma 17 m has tree-width at most 26g + 10a H . When considering the case that G does not contain K a,b as a subgraph, let S v be the set of vertices of G v that have at least a neighbors in G belonging to A v (and thus to A ′ v ); otherwise, let S v = ∅. Since there are at most t−1 a ways how to choose a set of a neighbors in A ′ v and no b vertices can have the same set of a neighbors in For v is planar, all vertices of β ′ ↑ v adjacent in G to a vertex of β ′ ↓ v belong to A ′ v (and thus there are at most t − 1 such vertices), and when considering the case that G does not contain K a,b as a subgraph, each vertex of β ′ ↓ v has at most a − 1 neighbors in G belonging to β ′ ↑ v.
Note that β ′ ↑ v can contain vertices not belonging to A ′ v , and thus β ′ ↑ v can have size larger than t − 1, and the tree decomposition (T, β ′ ) is not necessarily (t, t)-restricted. However, by the condition (e) from the statement of Theorem 16, the vertices of (β ′ ↑ v) \ A ′ v can only be contained in the bags of descendants of v which are disjoint from V (G v ), and thus we can fix up this issue as follows.
If w is a child of v and β ′ (w) ∩ V (G v ) = ∅, we say that the edge vw is skippable; note that in that case β ′ ↑ w ⊆ β ′ ↑ v. For each vertex w of T , let f (w) be the nearest ancestor of w such that the first edge on the path from f (w) to w in T is not skippable. Let T ′ be the rooted tree with vertex set V (T ) where the parent of each vertex w is f (w). Observe that (T ′ , β ′ ) is a tree decomposition of G[C]. Furthermore, denoting by z the child of f (w) on the path from f (w) to w in T , note that if a vertex x ∈ β ′ ↑ f (w) is contained in β ′ (w), then x ∈ β(z), and since the edge f (w)z is not skippable, the condition (e) from the statement of Theorem 16 implies that for each vertex v of T ′ , we conclude that (T ′ , β ′′ ) is a rooted tree decomposition of G[C] which is (t, t)restricted, and when considering the case that G does not contain K a,b as a subgraph, the decomposition is (t, a)-restricted.
Let us now consider the chromatic number of graphs with a (t, a)-restricted tree decomposition.
Lemma 19. Let a and t be positive integers. Let G be a graph with a (t, a)restricted rooted tree decomposition (T, β). The chromatic number of G is at most min(t + 3, a + 4). Additionally, if G is triangle-free, then the chromatic number of G is at most ⌈(13t + 172)/14⌉.

Proof.
We can color G using t + 3 colors, starting from the root of the tree decomposition, as follows. Suppose that we are considering a vertex v ∈ V (T ) such that β ↑ v is already colored. Since |β ↑ v| ≤ t − 1, this leaves at least 4 other colors to be used on G[β ↓ v]. Hence, we can extend the coloring to G[β ↓ v] by the Four Color Theorem.
We can also color G using a + 4 colors, starting from the root of the tree decomposition, as follows. For each vertex x of β ↓ v, at most a − 1 colors are used on its neighbors in β ↑ v, leaving x with at least 5 available colors not appearing on its neighbors. Since G[β ↓ v] is planar, we can color it from these lists of size at least 5 using the result of Thomassen [17], again extending the coloring to G[β ↓ v].
Finally, suppose that G is triangle-free. Let G ′ be the torso expansion of G with respect to (T, β), and let c = ⌈(13t+172)/14⌉. We again color G starting from the root of the tree decomposition using at most c colors. Additionally, we choose the coloring so that the following invariant is satisfied: (⋆) for each vertex w of T and for each independent set I in G[β(w)] such that I ∩ β ↓ w induces a clique in G ′ , at most c − 6 distinct colors are used on I.
Let v be a vertex of T . Suppose we have already colored β ↑ v, and we want to extend the coloring to β ↓ v. Note that the choice of this coloring may only affect the validity of the invariant (⋆) at v and at descendants of v in T . Consider any descendant w of T . Coloring β ↓ v can only assign color to vertices of β ↑ w, and since G ′ is the torso expansion of G, the set β ↑ w ∩ β ↓ v induces a clique in G ′ . Consequently, the validity of (⋆) at v implies the validity at w (until more vertices of G are assigned colors), and thus when choosing the coloring of β ↓ v, we only need to ensure that (⋆) holds at v.
The graph G ′ [β ↓ v] is planar, and thus it is 5-degenerate; i.e., there exists an ordering of its vertices such that each vertex is preceded by at most 5 of its neighbors. Let us color the vertices of β ↓ v according to this ordering, always preserving the validity of (⋆) at v. Suppose that we are choosing a color for a vertex x ∈ V (β ↓ v). Let P x consist of the neighbors of x in G ′ belonging to β ↓ v that precede it in the ordering; we have |P x | ≤ 5. Note that all cliques in G ′ [β ↓ v] containing x and with all other vertices already colored are subsets of P x ∪ {x}. Let Q x = P x ∪ β ↑ v; we have |Q x | ≤ t + 4.
Let N x consist of vertices of Q x that are adjacent to x in G. We say that a color a is forbidden at x if there exists an independent set A a ⊆ Q x \ N x of G such that A a ∩ P x is a clique in G ′ and c − 6 colors distinct from a appear on A a . Observe that assigning x a color which neither appears on N x nor is forbidden results in a proper coloring that preserves the invariant (⋆) at v.
Suppose first that no color is forbidden at x. Since G is triangle-free, N x \ P x is an independent set in G[β ↑ v], and by (⋆), at most c − 6 colors appear on N x \ P x . Since |P x | ≤ 5, it follows that some color does not appear on N x , as required.
Hence, we can assume that some color is forbidden at x, and thus there exists an independent set Z 1 ⊆ Q x \ N x of size at least c − 6 such that vertices of Z 1 are assigned pairwise distinct colors. Since |Q x | ≤ t + 4, at most t+4−(c−6) = t+10−c of these colors appear at least twice on Q x , and thus there exists a set Z 2 ⊆ Z 1 of size at least c − 6 − (t + 10 − c) = 2c − t − 16 such that the color of each vertex of Z 2 appears exactly once on Q x (and thus does not appear on N x ). Let Z = Z 2 \ P x ; we have |Z| ≥ 2c − t − 21. We claim that not all colors appearing on Z are forbidden; since such colors do not appear on N x , we can use them to color x.
For contradiction, assume that colors of all vertices of Z are forbidden at x. Let Z = {z 1 , . . . , z m } for some m ≥ 2c − t − 21, and for a = 1, . . . , m, let a be the color of z a . Since a is forbidden at x, there exists an independent set A a ⊆ Q x \N x such that A a ∩P x is a clique in G ′ and c−6 colors distinct from a appear on A a . Note that A a ∪{z a } is not an independent set, as otherwise this set contradicts the invariant (⋆) at v. Hence, we can choose a neighbor f (a) of z a in A a . Since Z is an independent set, we have f (a) ∈ Z. Furthermore, we claim that the preimage in f of each vertex has size at most 6: if say f (z 1 ) = . . . = f (z 7 ) = y, then for i = 1, . . . , 7, the vertex z i would have a neighbor y in the independent set A 1 , and thus z 1 , . . . , z 7 ∈ A 1 ; however, the only appearance of colors 1, . . . , 7 in Q x is on the vertices z 1 , . . . , z 7 , and thus at most c − 7 colors would appear on A 1 . We conclude that |f (Z)| ≥ |Z|/6, and thus t + 4 ≥ |Q x | ≥ |Z| + |f (Z)| ≥ 7 6 |Z| ≥ 7 6 (2c − t − 21) ≥ (6t + 25)/6. This is a contradiction.
Lemma 22. Let a be an integer such that 1 ≤ a ≤ (1 − 1/m)n. Suppose A is a subset of V of size n such that |A \ V 1 | = a. Let S be the set of pairs with one end in A ∩ V 1 and the other end in A \ V 1 . Then the probability that S ∩ E(G) = ∅ is less than 6e − pn 2592m a .