Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor

Kawarabayashi, Ken-ichi; Toft, Bjarne

doi:10.1007/s00493-005-0019-1

Cited by 44 publications

(35 citation statements)

References 26 publications

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“…All cases |V (H)| ≥ 7 remain unsolved. The only known results for |V (H)| = 7 are that any 7-colorable graph has K 7 or K 4,4 as a minor [KT05] and has K 7 or K 3,5 as a minor [Kawb], whereas Hadwiger's conjecture suggests that the graph should always have a K 7 minor. The best general upper bound is that every H-minor-free graph has a vertex coloring with O(|V (H)| lg |V (H)|) colors, which follows immediately from bounds on the average degree of a vertex in an H-minor-free graph; see, e.g., [Kos84,Tho01].…”

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confidence: 99%

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Demaine¹,

Hajiaghayi²,

Kawarabayashi³

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into odd-H-minor-free graphs, on the one hand generalizing the central structural result from Graph Minor Theory, and on the other hand providing an algorithmic decomposition into two bounded-treewidth graphs, generalizing a similar result for minors. As one example of how these structural results conquer difficult problems, we obtain a polynomial-time 2-approximation for vertex coloring in odd-H-minor-free graphs, improving on the previous O(|V (H)|)-approximation for such graphs and generalizing the previous 2-approximation for H-minor-free graphs. The class of odd-H-minor-free graphs is a vast generalization of the wellstudied H-minor-free graph families and includes, for example, all bipartite graphs plus a bounded number of apices. Odd-H-minorfree graphs are particularly interesting from a structural graph theory perspective because they break away from the sparsity of Hminor-free graphs, permitting a quadratic number of edges.

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confidence: 99%

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Demaine¹,

Hajiaghayi²,

Kawarabayashi³

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In 1993, Robertson, Seymour and Thomas [14] proved that the case k = 6 also follows from the Four Color Theorem. The cases k ≥ 7 are open; for the case k = 7, a partial result in [9] is all that is known.…”

Section: Introductionmentioning

confidence: 99%

Immersing small complete graphs

et al. 2010

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Following in the spirit of the Hadwiger and Hajós conjectures, Abu-Khzam and Langston have conjectured that every k-chromatic graph contains an immersion of K k . They proved this for k ≤ 4. Much before that, Lescure and Meyniel [10] obtained a stronger result that included also the values k = 5 and 6, by proving that every simple graph of minimum degree k − 1 contains an immersion of K k . They noted that they also have a proof of the same result for k = 7 but have not published it due to the length of the proof. We give a simple proof of this result. This, in particular, proves the conjecture of Abu-Khzam and Langston for every k ≤ 7.

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“…So far, the conjecture is open for every t ≥ 7. For the case t = 7, Kawarabayashi and Toft [19] proved that any 7-chromatic graph has K 7 or K 4, 4 as a minor. Recently, Kawarabayashi [18] proved that any 7-chromatic graph has K 7 or K 3, 5 as a minor.…”

Section: Theorem 13 For Every Integer K There Is a Computable Consmentioning

confidence: 99%

5-coloring K_3,k-minor-free graphs

Kawarabayashi

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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A seminal result of Thomassen [40] says that there are only finitely many 6-color-critical graphs for the bounded genus graphs. This result is no longer true if we consider K 3,k -minor-free graphs. K 3,k -minorfree graphs are a significant generalization of boundedgenus graphs. They also contain infinitely many t-colorcritical graphs for all t with k + 2 ≥ t ≥ 4.Motivated by this fact, we first show that if G is 6-color-critical 4-connected K 3,k -minor-free graphs, then G has tree-width at most g(k) for some function g of k. This allows us to show the following algorithmic result, which is of independent interest.For a 4-connected graph G and any k, there is an O(n 3 ) algorithm to test whether or not G is 5-colorable.Note that testing the 3-colorability of K 3,kminor-free graphs is NP-complete, and testing the 4-colorability of them would require a significant generalization of the Four Color Theorem (because K 3,3 -minor-free graphs are essentially planar). Testing the 5-colorability of bounded genus graphs can be done in polynomial time, as shown by Thomassen [40]. Thus our result can be viewed as a strengthening of the bounded genus case.We then investigate minimal forbidden minors in non-5-colorable K 3,k -minor-free graphs. Such a graph may exist, as K 6 shows. However, we prove the following result.There is a computable constant f (k) such that every minimal forbidden minor in non-5-colorable K 3,k -minorfree graphs has at most f (k) vertices.Our proof of the above result implies the following algorithmic result, which is of independent interest.For a graph G and any k, there is an O(n 3 ) algorithm to output one of the following:1. a 5-coloring of G, or * National Institute of Informatics and JST ERATO Kawarabayashi Project, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan. Email address: k keniti@nii.ac.jp † Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research and by Mitsubishi Foundation 2. a K 3,k -minor of G, or 3. a minor R of G of order at most f (k) (f (k) comes from the above theorem) such that R does not have a K 3,k -minor nor is 5-colorable.Let us emphasis that the chromatic number in our main results does NOT depend on the graph K 3,k that we are going to exclude as a minor. This is a big contrast with the algorithmic result of Hadwiger's conjecture [21].

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Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor

Cited by 44 publications

References 26 publications

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Immersing small complete graphs

5-coloring K_3,k-minor-free graphs

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Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor

Cited by 44 publications

References 26 publications

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Immersing small complete graphs

5-coloring K3,k-minor-free graphs

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Product

Resources

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5-coloring K_3,k-minor-free graphs