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Some remarks on the odd hadwiger’s conjecture

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Abstract

We say that H has an odd complete minor of order at least 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.

Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable.

Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al.

We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.

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Author notes

  1. Zi-Xia Song

    Present address: Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

Authors and Affiliations

  1. Graduate School of Information Sciences (GSIS), Tohoku University, Aramaki aza Aoba 09 Aoba-ku Sendai, Miyagi, 980-8579, Japan

    Ken-ichi Kawarabayashi

  2. Department of Mathematics, The Ohio State University, Columbus, OH43210, USA

    Zi-Xia Song

Authors
  1. Ken-ichi Kawarabayashi
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  2. Zi-Xia Song
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Correspondence to Ken-ichi Kawarabayashi.

Additional information

Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by Sumitomo Foundation and by Inoue Research Award for Young Scientists.

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Kawarabayashi, Ki., Song, ZX. Some remarks on the odd hadwiger’s conjecture. Combinatorica 27, 429–438 (2007). https://doi.org/10.1007/s00493-007-2213-9

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  • DOI: https://doi.org/10.1007/s00493-007-2213-9

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