Nonconstructive tools for proving polynomial-time decidability
Journal of the ACM (JACM)
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Acyclic colorings of planar graphs
Discrete Mathematics
Precoloring extension. I: Interval graphs
Discrete Mathematics - Special volume (part 1) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs”
List colourings of planar graphs
Discrete Mathematics
Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Every planar graph is 5-choosable
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Scheduling with incompatible jobs
Discrete Applied Mathematics
Parallel complexity of partitioning a planar graph into vertex-induced forests
Discrete Applied Mathematics
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
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
Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Efficient algorithms for acyclic colorings of graphs
Theoretical Computer Science
Journal of Combinatorial Theory Series B
Computing vertex connectivity: new bounds from old techniques
Journal of Algorithms
Surfaces, tree-width, clique-minors, and partitions
Journal of Combinatorial Theory Series B
The extremal function for complete minors
Journal of Combinatorial Theory Series B
A characterization of weakly bipartite graphs
Journal of Combinatorial Theory Series B
A short proof of Guenin's characterization of weakly bipartite graphs
Journal of Combinatorial Theory Series B
Packing odd circuits in Eulerian graphs
Journal of Combinatorial Theory Series B
Embedding k-outerplanar graphs into ℓ1
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
NC Algorithms for Partitioning Sparse Graphs into Induced Forests with an Application
ISAAC '95 Proceedings of the 6th International Symposium on Algorithms and Computation
Efficient Algorithms for Vertex Arboricity of Planar Graphs
Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science
Dimension Reduction in the \ell _1 Norm
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Cuts, Trees and -Embeddings of Graphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Implementation of algorithms for maximum matching on nonbipartite graphs.
Implementation of algorithms for maximum matching on nonbipartite graphs.
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
An improved linear edge bound for graph linkages
European Journal of Combinatorics - Special issue: Topological graph theory II
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
On Sufficient Degree Conditions for a Graph to be $k$-linked
Combinatorics, Probability and 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
Non-zero disjoint cycles in highly connected group labelled graphs
Journal of Combinatorial Theory Series B
Some remarks on the odd hadwiger’s conjecture
Combinatorica
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Combinatorica
A relaxed Hadwiger's Conjecture for list colorings
Journal of Combinatorial Theory Series B
A nearly linear time algorithm for the half integral disjoint paths packing
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Additive approximation algorithms for list-coloring minor-closed class of graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
List-coloring graphs without K4,k-minors
Discrete Applied Mathematics
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Hadwiger's conjecture is decidable
Proceedings of the forty-first annual ACM symposium on Theory of computing
Note: Note on coloring graphs without odd-Kk-minors
Journal of Combinatorial Theory Series B
Decomposition, approximation, and coloring of odd-minor-free graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Faster approximation schemes and parameterized algorithms on H-minor-free and odd-minor-free graphs
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
List-coloring graphs without subdivisions and without immersions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs
Theoretical Computer Science
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On the excluded minor structure theorem for graphs of large tree-width
Journal of Combinatorial Theory Series B
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It is well-known (Feige and Kilian [24], Håstad [39]) that approximating the chromatic number within a factor of n1-ε cannot be done in polynomial time for ε0, unless coRP = NP. Computing the list-chromatic number is much harder than determining the chromatic number. It is known that the problem of deciding if the list-chromatic number is k, where k ≥ 3, is Π2p-complete [37].In this paper, we focus on minor-closed and odd-minor-closed families of graphs. In doing that, we may as well consider only graphs without Kk-minors and graphs without odd Kk-minors for a fixed value of k, respectively. Our main results are that there is a polynomial time approximation algorithm for the list-chromatic number of graphs without Kk-minors and there is a polynomial time approximation algorithm for the chromatic number of graphs without odd-Kk-minors. Their time complexity is O(n3) and O(n4), respectively. The algorithms have multiplicative error O(√log k) and additive error O(k), and the multiplicative error occurs only for graphs whose list-chromatic number and chromatic number are Θ(k), respectively.Let us recall 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 are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Let us observe that the complete bipartite graph Kn/2,n/2 contains a Kk-minor for k ≤ n/2, but on the other hand, it does not contain an odd Kk-minor for any k ≥ 3. Odd K5-minor-free graphs are closely related to one field of discrete optimization which is finding conditions under which a given polyhedron has integer vertices, so that integer optimization problems can be solved as linear programs. See [33, 34, 64]. Also, the odd version of the well-known Hadwiger's conjecture has been considered, see [28].Our main idea involves precoloring extension. This idea is used in many results; one example is Thomassen's proof on his celebrated theorem on planar graphs [69].The best previously known approximation for the first result is a simple O(k √log k)-approximation following algorithm that guarantees a list-coloring with O(k √log k) colors for Kk-minor-free graphs. This follows from results of Kostochka [54, 53] and Thomason [67, 68].The best previous approximation for the second result comes from the recent result of Geelen et al. [28] who gave an O(k √log k)-approximation algorithm.We also relate our algorithm to the well-known conjecture of Hadwiger [38] and its odd version. In fact, we give an O(n3) algorithm to decide whether or not a weaker version of Hadwiger's conjecture is true. Here, by a weaker version of Hadwiger's conjecture, we mean a conjecture which says that any 27k-chromatic graph contains a Kk-minor. Also, we shall give an O(n2500k) algorithm for deciding whether or not any 2500k-chromatic graph contains an odd-Kk-minor.Let us mention that this presentation consists of two papers which are merged into this one. The first one consists of results concerning minor-closed classes of graphs by two current authors, and the other consists of results concerning odd-minor-closed classes of graphs by the first author.