Graph minors. V. Excluding a planar graph
Journal of Combinatorial Theory Series B
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Typical subgraphs of 3- and 4-connected graphs
Journal of Combinatorial Theory Series B
Every planar graph is 5-choosable
Journal of Combinatorial Theory Series B
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Highly connected sets and the excluded grid theorem
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
Depth-First Search and Kuratowski Subgraphs
Journal of the ACM (JACM)
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor
Combinatorica
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture
Journal of Combinatorial Theory Series B
A relaxed Hadwiger's Conjecture for list colorings
Journal of Combinatorial Theory Series B
Locally planar graphs are 5-choosable
Journal of Combinatorial Theory Series B
List-color-critical graphs on a fixed surface
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Paired approximation problems and incompatible inapproximabilities
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Fast minor testing in planar graphs
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Faster parameterized algorithms for minor containment
Theoretical Computer Science
Finite dualities and map-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B
List-coloring graphs without subdivisions and without immersions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Faster parameterized algorithms for minor containment
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
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The famous Hadwiger's conjecture asserts that every graph with no Kt-minor is (t-1)-colorable. The case t=5 is known to be equivalent to the Four Color Theorem by Wagner, and the case t=6 is settled by Robertson, Seymour and Thomas. So far the cases t ≥ 7 are wide open. In this paper, we prove the following two theorems: There is an O(n2) algorithm to decide whether or not a given graph G satisfies Hadwiger's conjecture for the case t. Every minimal counterexample to Hadwiger's conjecture for the case t has at most f(t) vertices for some explicit bound f(t). The bound f(t) is at most pppt, where p=101010t. Our proofs for both results use the well-known result by Thomassen [46] for 5-list-coloring planar graphs, together with some results (but not the decomposition theorem) of Graph Minors in [36]. Concerning the first result, we prove the following stronger theorem: For a given graph G and any fixed t, there is an O(n2) algorithm to output one of the following: a (t-1)-coloring of G, or a Kt-minor of G, or a minor H of G of order at most f(t) such that H does not have a Kt-minor nor is (t-1)-colorable. The last conclusion implies that H is a counterexample to Hadwiger's conjecture with at most f(t) vertices for the case t. The time complexity of the algorithm matches the best known algorithms for 4-coloring planar graphs (the Four Color Theorem), due to Appel and Hakken, and Robertson, Sanders, Seymour and Thomas, respectively. Let us observe that when t=5, the algorithm gives rise to an algorithm for the Four Color Theorem. The second theorem follows from our structure theorem, which has the following corollary: Every minimal counterexample G to Hadwiger's conjecture for the case t either has at most f(t) vertices, or has a vertex set Z of order at most t-5 such that G-Z is planar. It follows from the Four Color Theorem that the second assertion does not happen to any minimal counterexample to Hadwiger's conjecture for the case t. Thus in constant time, we can decide Hadwiger's conjecture for the case t.