On the odd-minor variant of Hadwiger’s conjecture Jim Geelen∗, Bert Gerards†, Bruce Reed‡, Paul Seymour§ Adrian Vetta¶ ???; revised April 18, 2011 ∗Department of Combinatorics and Optimization, University of Waterloo. Email:
[email protected] †Centrum voor Wiskunde en Informatica, and Eindhoven University of Technology. Email:
[email protected] ‡School of Computer Science, McGill University. Email:
[email protected] §Department of Mathematics, Princeton University. Email:
[email protected] ¶Department of Mathematics and Statistics, and School of Computer Science, McGill University. Email:
[email protected] Abstract A Kl-expansion consists of l vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd Kl-expansion then its chromatic number is O(l√log l). In doing so, we obtain a characterization of graphs which contain no odd Kl-expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k 2 vertices such that every odd path − with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results. 1 The Main Results All graphs in this paper are finite, and have no loops or parallel edges.