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Journal Articles Journal of Combinatorial Theory, Series B Year : 2009

On the odd-minor variant of Hadwiger's conjecture


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 View the MathML source. 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 2kâˆ'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.
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Dates and versions

hal-00795995 , version 1 (07-03-2013)


  • HAL Id : hal-00795995 , version 1


Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, Adrian Vetta. On the odd-minor variant of Hadwiger's conjecture. Journal of Combinatorial Theory, Series B, 2009, 99, pp.20―29. ⟨hal-00795995⟩
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