On the odd-minor variant of Hadwiger's conjecture

J.F. Geelen, B. Gerards, B. Reed, P.D. Seymour, A. Vetta

Research output: Contribution to journalArticleAcademicpeer-review

35 Citations (Scopus)
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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 . 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.
Original languageEnglish
Pages (from-to)20-29
Number of pages13
JournalJournal of Combinatorial Theory, Series B
Volume99
Issue number1
DOIs
Publication statusPublished - 2009

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