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 n1-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 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.