Approximating the list-chromatic number and the chromatic number in minor-closed and odd-minor-closed classes of graphs

It is well-known (Feige and Kilian [24], Håstad [39]) that approximating the chromatic number within a factor of n^1-ε cannot be done in polynomial time for ε>0, 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 Π₂^p-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 K_k-minors and graphs without odd K_k-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 K_k-minors and there is a polynomial time approximation algorithm for the chromatic number of graphs without odd-K_k-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 K_n/2,n/2 contains a K_k-minor for k ≤ n/2, but on the other hand, it does not contain an odd K_k-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 K_k-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 K_k-minor. Also, we shall give an O(n^2500k) algorithm for deciding whether or not any 2500k-chromatic graph contains an odd-K_k-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.