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Discrete Applied Mathematics 157 (2009) 659–662
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
List-coloring graphs without K4,k-minorsKen-ichi KawarabayashiThe National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
a r t i c l e i n f o
Article history:Received 26 April 2005Received in revised form 22 July 2008Accepted 7 August 2008Available online 9 September 2008
Keywords:List-coloringHadwiger’s conjecture
a b s t r a c t
In this note, it is shown that every graph with no K4,k-minor is 4k-list-colorable. We alsogive an extremal function for the existence for a K4,k-minor. Our proof implies that there isa linear time algorithm for showing that either G has a K4,k-minor or G is 4k-choosable. Infact, if the latter holds, then the algorithm gives rise to a 4k-list-coloring.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
In this paper, all graphs are finite and simple. We follow standard graph theory terminology and notation as used, forexample, in [4]. A graph H is aminor of a graph K if H can be obtained from a subgraph of K by contracting edges.Our research is motivated by Hadwiger’s conjecture from 1943 which suggests a far reaching generalization of the Four-
Color Theorem, and it is perhaps one of the most challenging open problems in graph theory.
Conjecture 1.1 (Hadwiger [6]). For all k ≥ 1, every k-chromatic graph has the complete graph Kk on k vertices as a minor.
For k = 1, 2, 3, it is easy to prove, and for k = 4, Hadwiger himself [6] and Dirac [5] proved it. For k = 5, however, it seemsextremely difficult. In 1937, Wagner [29] proved that the case k = 5 is equivalent to the Four-Color Theorem. So, assumingthe Four-Color Theorem, the case k = 5 of Hadwiger’s conjecture holds. In 1993, Robertson, Seymour and Thomas [23]proved that a minimal counterexample to the case k = 6 is a graph Gwhich has a vertex v such that G− v is planar. By theFour-Color Theorem, the case k = 6 of Hadwiger’s conjecture holds. This result is the deepest in this research area. Hencethe cases k = 5, 6 are each equivalent to the Four-Color Theorem [1,2,22]. So far, the conjecture is open for every k ≥ 7.For the case k = 7, Toft and the author [14] proved that any 7-chromatic graph has K7 or K4, 4 as a minor. Recently, theauthor [10] proved that any 7-chromatic graph has K7 or K3, 5 as a minor.It is even not known whether there exists an absolute constant c such that every ck-chromatic graph has a Kk-minor. So
far, it is known that there exists a constant c such that every ck√log k-chromatic graph has a Kk-minor. This follows from the
results of Kostochka [17,16] or Thomason [24,25]. This was proved 25 years ago, but nobody can improve the superlinearorder k
√log k. So it would be of great interest to prove that a linear function of the chromatic number is sufficient to force a
Kk-minor. From an algorithmic view, we can ‘‘decide’’ this problem in polynomial time. This was proved in [11,13]. We referthe reader to [27] for further information on Hadwiger’s conjecture.Let G be a graph. A list-assignment is a function L which assigns to every vertex v ∈ V (G) a set L(v) of natural numbers,
which are called admissible colors for that vertex. An L-coloring of the graph G is an assignment of admissible colors to allvertices of G, i.e., a function c : V (G)→ N such that c(v) ∈ L(v) for every v ∈ V (G), and c(u) 6= c(v) for every edge uv. If kis an integer and |L(v)| ≥ k for every v ∈ V (G), then L is a k-list-assignment. A graph is k-choosable if it admits an L-coloringfor every k-list-assignment L.
E-mail address: [email protected].
0166-218X/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.dam.2008.08.015
660 K.-i. Kawarabayashi / Discrete Applied Mathematics 157 (2009) 659–662
When relaxing Hadwiger’s conjecture to allow ck colors, the following conjecture involving list-colorings may also betrue.
Conjecture 1.2. There exists a constant c such that every graph without Kk-minors is ck-choosable.
This was conjectured in [12]. Also, the following weaker version of the choosability analog of Hadwiger’s conjecture wasgiven by Woodall [32,33].
Conjecture 1.3 (Woodall [32]). Every graph with no Kr,s-minor is (r + s− 1)-choosable.
Note that the choosability analog of Hadwiger’s conjecture is false, because there is a planar graph which is not 4-choosable [28] (but every planar graph is 5-choosable [26]).Again, it is even not known that Conjecture 1.2 holds, so it would be of interest to know even the cases where r is small.
For the usual graph coloring,Woodall [31] made a related conjecture.Woodall [32,33] proved Conjecture 1.3 when r = 1, 2.The main purpose of this note concerns the cases r = 3, 4. We shall prove the following result.
Theorem 1.4. If G does not contain a K4,k-minor, then G is 4k-choosable.
This immediately implies that every graph without a K3,k-minor is 4k-choosable.Actually, Theorem 1.4 follows from the following extremal function for the existence of a K4,k-minor, which is of
independent interest.
Theorem 1.5. Let G be a graph such that |V (G)| ≥ 2k + 2 and |E(G)| ≥ 2k(|V (G)| − k − 1) + 1, where k ≥ 2. Then G has aK4,k-minor.
Let us remark that the extremal function ‘‘2k(|V (G)| − k − 1) + 1’’ is perhaps not best possible, and we conjecturethat the factor 2 is not necessary. If true, this, together with our proof, would imply that every graph with no K4,k-minor is2k-choosable. Let us observe that the cases k ≤ 4 are already proved in [7], and are best possible.There are several results [3,15,18–20] concerning the extremal function for the existence of complete bipartite graph
minors. These results say that if k is large enough compared to s, then an average degree just over k suffices to ensure theexistence of a Ks,k-minor. In particular, the result in [15] implies that if k is sufficiently large, then every graph with no K4,k-minor is (k+ 13)-choosable. But these results give no information on the chromatic number of a graph with no K4,k-minorwhen k is not sufficiently large. Hence it is useful to have our result, which applies to every graph with no K4,k-minor. Notethat when k = 4, the result in [7] implies that every graph with no K4,4-minor is 8-choosable.To see that Theorem 1.5 implies Theorem 1.4, consider a minimal counterexample G to Theorem 1.4. When k = 1, then
the result easily holds. So suppose k ≥ 2. Then it is easy to prove that the minimum degree is at least 4k. For suppose notand that v has degree at most 4k − 1. Then, by induction, G − v has a desired list-coloring. Since v has at least 4k colorsavailable, we can easily extend the list-coloring of G − v to G. Hence every vertex has degree at least 4k, and this impliesthat G has at least 4k+ 1 vertices and at least 2k|V (G)| edges. On the other hand, by Theorem 1.5, if G has at least 2k|V (G)|edges for k ≥ 2 and |V (G)| ≥ 2k + 2, then G has a K4,k-minor. Thus we can conclude that G is not a counterexample, andhence Theorem 1.4 holds.Let us point out that this proof implies a polynomial time algorithm to 4k-list-color a graph with no K4,k-minor. More
precisely, there is a linear time algorithm (linear in the number of vertices plus the number of edges of G) which showseither
(1) G is 4k-choosable, or(2) G has a K4,k-minor.
Actually, when (1) holds, the algorithm gives a 4k-list-coloring. To see this, if G has a vertex v of degree at most 4k− 1,thenwe just delete v from G. We keep doing this procedure until there is no vertex of degree atmost 4k−1. If this procedurecontinues until the resulting graph is empty, then clearly we can 4k-list-color G recursively in linear time. If the resultinggraph is not empty, this graph contains a K4,k-minor by Theorem 1.5. This can be clearly found in linear time.In order to prove our main result, we need the definition of ‘‘rooted minor’’.If v1, . . . , vk are k distinct vertices in a graph G, then we say that G has a Ka,k-minor rooted at v1, . . . , vk if there are
disjoint connected subgraphs H1, . . . ,Ha, K1, . . . , Kk such that each Ki contains vi and is adjacent to all of H1, . . . ,Ha. Wesay that G has every rooted Ka,k-minor if, for every choice of k distinct vertices v1, . . . , vk of G, G has a Ka,k-minor rooted atv1, . . . , vk.Rooted minor problems are studied by many researchers. For example, the following result is known.
Theorem 1.6 (Robertson–Seymour [21]). Suppose G is 3-connected. Then G has every rooted K2,3-minor unless G is planar, inwhich case it has a K2,3-minor rooted at v1, v2, v3 if and only if v1, v2, v3 are not all on the boundary of the same face.
K.-i. Kawarabayashi / Discrete Applied Mathematics 157 (2009) 659–662 661
Theorem 1.6 follows from (2.4) in [21]. See also (3.5) in [23].Actually, this result is used to prove more interesting results on Hadwiger’s conjecture; see [14,23].Furthermore, every rooted K2,4-minor problem and every rooted K3,4-minor problem are discussed in [7,8], respectively.
Moreover, the general connectivity function for the existence of every rooted minor is obtained by the author [9].Recently, the following extremal function for the existence of every rooted K2,k-minor was obtained by Wollan [30].
Theorem 1.7. Every k-connected graph G on n vertices with at least kn−(k+12
)+ 1 edges contains every rooted K2,k-minor.
The lower bound on the number of edges in Theorem 1.7 is best possible. To see this, let G be the join of a maximal planargraphH of order n−k+3, and a complete graphKk−3. Then |E(G)| = 3(n−k+3)−6+
(k−32
)+(n−k+3)(k−3) = kn−
(k+12
).
If v1, v2, v3 are in one face of H and v4, . . . , vk are the vertices of Kk−3, then just as in Theorem 1.6, there is no K2,k-minorrooted at v1, . . . , vk.We will use Theorem 1.7 in our main proof.
2. Proof of the main theorem
We prove Theorem 1.5 by induction on the number of vertices. We first prove the case |V (G)| = 2k + 2. Then it is easyto see that |E(G)| ≥
(2k+22
)− k. Therefore, there are at most k missing edges in G. Suppose k ≥ 4. Then clearly there are
four vertices that have at least n− 6 = 2k− 4 ≥ k common neighbors. Therefore, G has K4,k as a subgraph. If k ≤ 3, then Ghas at most eight vertices. Since there are at most three missing edges, we can easily find K4,k as a subgraph in G.Thus we may assume |V (G)| ≥ 2k+ 3. If G has a vertex v of degree at most 2k, then|E(G− v)| ≥ |E(G)| − 2k
≥ 2k(n− k− 1)+ 1− 2k= 2k(|V (G− v)| − k− 1)+ 1.
Thus by induction, G− v has a K4,k-minor. So, we may assume that every vertex in G has degree at least 2k+ 1.We now claim that G is (k+2)-connected. Suppose not. Then G has subgraphs A and B such that G = A∪B, both A−B and
B−A are not empty, and |V (A)∩V (B)| ≤ k+1. Since every vertex has degree at least 2k+1, it follows that |A|, |B| ≥ 2k+2.Wemay assume that neither A nor B satisfies the induction hypothesis of Theorem 1.5, since otherwise there is a K4,k-minorin either A or B by the induction hypothesis. Then
|E(G)| ≤ |E(A)| + |E(B)|≤ 2k(|V (A)| − k− 1)+ 2k(|V (B)| − k− 1)≤ 2k(|V (A)| + |V (B)| − 2k− 2)≤ 2k(|V (G)| + k+ 1− 2k− 2)≤ 2k(|V (G)| − k− 1).
This contradicts the hypothesis of Theorem 1.5. Thus G is (k+ 2)-connected, as claimed.We also claim that every edge in E(G) is contained in at least 2k triangles. Suppose not, and there is an edge e ∈ E(G)
such that e is contained in at most 2k−1 triangles. Let G′ be the graph obtained from G by contracting e. Then it follows that|E(G′)| ≥ |E(G)| − 2k
≥ 2k(|V (G′)| − k− 1)+ 1.Thus G′ satisfies the induction hypothesis of Theorem 1.5, and so G′ contains a K4,k-minor. Hence wemay assume that everyedge is contained in at least 2k triangles.Let e = xy be any edge of E(G), and let v1, . . . , vk be k vertices that are adjacent to both x and y. Let G′′ = G − {x, y}.
Clearly G′′ is k-connected since G is (k + 2)-connected. Note that G′′ omits at most 2|V (G)| − 3 edges of G. Suppose k ≥ 3.Since G has at least 2k+ 3 vertices,
|E(G′′)| ≥ |E(G)| − 2|V (G)| + 3≥ 2k(|V (G)| − k− 1)+ 1− 2|V (G)| + 3= k(|V (G)| − 2)+ (k− 2)|V (G)| − 2k2 + 4≥ k(|V (G)| − 2)− k− 2
≥ k|V (G′′)| −(k+ 12
)+ 1.
Thus G′′ has a K2,k-minor rooted at v1, . . . , vk by Theorem 1.7. Together with x, y, this gives rise to a K4,k-minor.It remains to consider the case k = 2. IfG is a complete graph, clearly it contains K4,2 as a subgraph. Take two nonadjacent
vertices x′, y′. Since G is now 4-connected, there are four disjoint paths between x′ and y′, and clearly this gives rise to a K4,2-minor.This completes the proof of Theorem 1.5.
662 K.-i. Kawarabayashi / Discrete Applied Mathematics 157 (2009) 659–662
Acknowledgments
I would like to thank the referee who suggested improvement of both the result and the proof. I also thank Paul Wollanfor letting me know his result from [30]. The author’s research was partly supported by the Japan Society for the Promotionof Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by the Sumitomo Foundation, by the C & CFoundation, by the Inamori Foundation, and by an Inoue Research Award for Young Scientists.
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