Journal of Combinatorial Theory, Series B 99 (2009) 728–731
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Note
Note on coloring graphs without odd-K k -minors ✩ Ken-ichi Kawarabayashi National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
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Article history: Received 10 March 2008 Available online 13 December 2008 Keywords: Hadwiger’s conjecture Odd minor
We give a short proof that every graph G without an odd K k -minor is O (k log k )-colorable. This was first proved by Geelen et al. [J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the oddminor variant of Hadwiger’s conjecture, J. Combin. Theory Ser. B 99 (1) (2009) 20–29]. We give a considerably simpler and shorter proof following their approach. © 2008 Elsevier Inc. All rights reserved.
Hadwiger’s conjecture from 1943 [8] suggests a far-reaching generalization of the Four Color Theorem [1,2,18] and is considered by many to be one of the deepest open problems in graph theory. Hadwiger’s conjecture says that any graph without K k as a minor is (k − 1)-colorable. In 1937, Wagner [23] proved that the case k = 5 of the conjecture is, in fact, equivalent to the Four Color Theorem. In 1993, Robertson, Seymour and Thomas [20] proved that the case k = 6 also follows from the Four Color Theorem. The cases k 7 are open, and even for the case k = 7,the partial result in [12] is best known. It is known that any graph G without K k as a minor is O (k log k )-colorable, see [15,16,21, 22]. Currently, this result is best known for the general case. Recently, the concept “odd-minor” has drawn attention by many researchers, because of its relation to Graph Minor Theory, maximum cut problem and Hadwiger’s conjecture. We say that H has an odd complete minor of order l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees can be two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Hence it is easy to see that odd minor is a generalization of minor. Concerning the connection to Hadwiger’s conjecture, Gerards and Seymour (see [9, p. 115]) conjectured that any graph without an odd K k -minor is (k − 1)-colorable. The k = 4 case was proved by Catlin [4]. Recently, Guenin [7] announced a solution of the k = 5 case. The main purpose of this note is to prove the following result.
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Research partially supported by JSPS Postdoctoral Fellowship for Research Abroad. E-mail address:
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0095-8956/$ – see front matter doi:10.1016/j.jctb.2008.12.001
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Elsevier Inc. All rights reserved.
K.-i. Kawarabayashi / Journal of Combinatorial Theory, Series B 99 (2009) 728–731
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Theorem 1. Any graph with no odd K k -minor is O (k log k )-colorable. Theorem 1 is clearly a generalization of the result by Kostochka and Thomason [15,16,21,22]. This is first proved by Geelen et al. [6]. We follow their approach, and find a short cut which makes a proof considerably simpler and shorter. So, this note may be viewed as supplement and appendix to the paper [6]. Let us point out that the paper [6] has some applications. For instance, the paper [14] extends the connectivity result for the existence of K k -minors [3] to that for the existence of odd K k -minors by using a result in [6]. Other applications may be found in [11,13]. Our proof is based on induction on the number of vertices. For the induction purpose, we shall prove the following strongerstatement. Suppose that c 1 is a constant such that any graph with minimum degree at least ck log k contains a K k -minor. Theorem 2. For any vertex set Z in G with | Z | 4k, either G has an odd K k -minor or any precoloring of the subgraph of G induced by Z can be extended to a (64ck log k + 4k)-coloring of G. Proof. If V (G ) = Z , there is nothing to prove. Let G be a minimal counterexample with |G | minimum. Suppose G − Z has a vertex v of degree at most 64ck log k + 4k − 1. Then, by the minimality of G, G − v has a (64ck log k + 4k)-coloring. Clearly this coloring can be extended to a coloring of G, a contradiction. So, we may assume that every vertex v ∈ V (G − Z ) has degree at least (64ck log k + 4k). Let A and B be vertex sets of G such that G = G [ A ] ∪ G [ B ]. Then we say that the pair ( A , B ) is a separation of G. The order of the separation is equal to | A ∩ B |. We shall prove the following: (1) There is no separation ( A , B ) of order at most 2k in G such that both A − B − Z and B − A − Z are nonempty. Suppose that there is such a separation ( A , B ) of order at most 2k. Let S = A ∩ B. Since | S | 2k and | Z | 4k, it follows that either | S ∪ ( A ∩ Z )| 4k or | S ∪ ( B ∩ Z )| 4k, say | S ∪ ( A ∩ Z )| 4k. By the minimality of G, we can get a coloring of the subgraph of G induced on B ∪ Z with Z precolored. Note that subgraph of G induced on B ∪ Z is smaller than G. Again, by the minimality of G, we can color G [ A ] with Z = S ∪ ( A ∩ Z ) precolored, where the precoloring of vertices in S comes from the coloring of G [ B ∪ Z ]. Recall that | Z | 4k and the subgraph induced on A is smaller than G. So we have a desired coloring of G [ A ] with Z precolored. Finally, the combination of the obtained colorings of G [ B ] and G [ A ] yields a desired coloring of G, a contradiction. We choose a spanning bipartite graph H of G − Z such that minimum degree of H is as large as ˝ says that every graph of minimum degree at least 2l has a spanning possible. An old theorem of Erdos bipartite graph of minimum degree at least l. Since G − Z has minimum degree at least 64ck log k, H has minimum degree at least 32ck log k. By Mader’s theorem [17], H has the following highly connected subgraph.
(2) H has an 8ck log k-connected subgraph L. We say that P is a parity breaking path for L if P is disjoint from L except for its endpoints, and L together with P has an odd cycle. We shall use the following recent result in [5]. (3) For any set S of vertices of a graph G, either 1. there are k disjoint odd S paths, i.e., k disjoint paths each of which has an odd number of edges and both its endpoints in S, or 2. there is a vertex set X of order at most 2k − 2 such that G − X contains no such paths.
Since L is 8ck log k-connected, each of the partite sets of L has at least 8ck log k vertices. (4) There are at least k disjoint parity breaking paths for L.
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Take one partite set S of L. We shall apply (3) to G and S. If there are at least k vertex-disjoint odd S paths in G, we can clearly find k vertex-disjoint parity breaking paths for L. Otherwise, by (3), there is a vertex separation ( A , B ) of order at most 2k − 2 such that G [ A ] contains S and G [ A − B ] has no such paths. Since | A ∩ B | 2k − 2 and L is 8ck log k-connected, it follows that G [ A ] contains L. By (1), B − A − Z = ∅. It follows from (3) that G [ A − B ] can be expressed as L ∪ W 1 ∪ W 2 ∪ · · · ∪ W m for some integer m such that L is a bipartite graph containing L, and each W i is a block. In addition, each W i contains a cutvertex v i to L such that W i − v i consists of vertices in Z (1 i m). Otherwise, there would be a separation ( A , B ) of order at most 2k such that both A − B − Z and B − A − Z are nonempty, a contradiction to (1). Since | Z | 4k, hence m 4k. Because B − A − Z = ∅, we need at most 4k colors for Z , another at most 4k colors for the vertices v 1 , . . . , v m , 2 colors for L , and at most 2k − 2 colors for A ∩ B, since | A ∩ B | 2k − 2. Clearly these colors are enough to color G, and we only need at most 10k 8ck log k + 4k colors for the coloring of G. This completes the proof of (4). The next lemma is proved in [10]. (5) Let G be a graph and S = {s1 , . . . , sk } be a set of k vertices. Suppose G has a K 2k -minor and is k-connected. Then G has k vertex disjoint nonempty connected subgraphs C 1 , . . . , C k such that, for each 1 i k, the subgraph C i contains si and is adjacent to all the subgraphs C 1 , . . . , C i −1 , C i +1 , . . . , C k . Actually, the statement in [10] is slightly different, but the proof in [10] or even the proof in [19] does imply (5). Let P 1 , . . . , P k be the parity breaking paths for L with endpoints { p i , p i } for 1 i k. By our choice of c, L has a K 4k -minor. So (4) and (5) imply the following: (6) G has a K 2k -minor made up of pairwise disjoint trees N 1 , . . . , N 2k , each of which is contained in the bipartite graph L, and for which there are k vertex disjoint parity breaking paths P 1 , . . . , P k such that P i has one endpoint p i in N 2i −1 and the other p i in N 2i , but is otherwise disjoint from this clique minor, which is bipartite. The induced subgraph consisting of trees N 1 , . . . , N 2k in L is clearly bipartite. Let us construct an odd-K k -minor from (6). Suppose we can two-color N j ∪ P j ∪ N j +k so that trees N i ∪ P i ∪ N i +k consist of an odd K j -minor for 1 i j. We claim that we can two color N j +1 ∪ P j +1 ∪ N j +1+k so that trees N i ∪ P i ∪ N i +k consist of an odd K j +1 -minor for 1 i j + 1. This is certainly possible since the two-coloring of N j +1+k can be swapped by the two-coloring of N j +1 in a sense of bipartition of L. So for any i j, either the edge between N i and N j +1 or the edge between N i +k and N j +1+k is monochromatic. In this way, we can construct an odd K k -minor. This completes the proof. 2 Acknowledgments I thank Bruce Reed and Paul Seymour for sending me the manuscript of [6]. I also thank the referee for suggesting improvements of the presentation. References [1] K. Appel, W. Haken, Every planar map is four colorable, part I: Discharging, Illinois J. Math. 21 (1977) 429–490. [2] K. Appel, W. Haken, J. Koch, Every planar map is four colorable, part II: Reducibility, Illinois J. Math. 21 (1977) 491–567. [3] T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite minors, J. Combin. Theory Ser. B 99 (3) (2009) 557–582. [4] P.A. Catlin, A bound on the chromatic number of a graph, Discrete Math. 22 (1978) 81–83. [5] M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman, P. Seymour, Packing non-zero A-paths in group labelled graphs, Combinatorica 26 (2006) 521–532. [6] J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the odd-minor variant of Hadwiger’s conjecture, J. Combin. Theory Ser. B 99 (1) (2009) 20–29. [7] B. Guenin, Talk at Oberwolfach on graph theory, January 2005. [8] H. Hadwiger, Über eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch. Ges. Zürich 88 (1943) 133–142.
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