The odd Hadwiger's conjecture is "almost" decidable

The concept “odd-minor” which is a generalization of minor-relation has received considerable amount of attention by many researchers, and led to several beautiful conjectures and results. We say 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. Hence it is easy to see that odd minor is a generalization of minor. Let us observe that the complete bipartite graph Kn/2,n/2 certainly contains a Kk-minor for k ≤ n/2, but on the other hand, it does not contain any odd Kk-minor for any k ≥ 3. So odd-minor-closed graphs seem to be much weaker than minor-closed graphs. The odd Hadwiger’s conjecture, made by Gerads and Seymour in early 1990s, is an analogue of the famous Hadwiger’s conjecture. It says that every graph with no odd Kt-minor is (t− 1)-colorable. This conjecture is known to be true for t ≤ 5, but the cases t ≥ 5 are wide open. So far, the most general result says that every graph with no odd Kt-minor is O(t √ log t)-colorable. In this paper, we tackle this conjecture from an algorithmic view, and show the following: For a given graph G and any fixed t, there is a polynomial time algorithm to output one of the following: 1. a (t− 1)-coloring of G, or 2. an odd Kt-minor of G, or 3. after making all “reductions” to G, the resulting graph H (which is an odd minor of G and which has no reductions) has a tree-decomposition (T, Y ) such that torso of each bag Yt is either • of size at most f1(t) log n for some function f1 of t, or • a graph that has a vertex X of order at most f2(t) for some function f2 of t such that Yt−X is bipartite. Moreover, degree of t in T is at most f3(t) for some function f3 of t. Let us observe that the last odd minorH is indeed a minimal counterexample to the odd Hadwiger’s conjecture for the case t. From this we obtain the following: For a given graph G and any fixed t, there is a polynomial time algorithm to output one of the following: 1. a (t− 1)-coloring of G, or 2. an odd Kt-minor of G, or 3. after making all “reductions” to G, we can color the resulting graph H (which is an odd minor of G and which has no reductions) with at most χ(H)+1 colors in polynomial time, where χ(H) is the chromatic number of H . In the last conclusion, we can actually figure out whether or not H contains an odd Kt-minor. 17 May 2015, revised .