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Connected matchings and Hadwiger's conjecture

Published online by Cambridge University Press:  11 April 2005

ZOLTÁN FÜREDI ,
ANDRÁS GYÁRFÁS  and
GÁBOR SIMONYI
ZOLTÁN FÜREDI
Affiliation:
Alfréd Rényi Institute, Hungarian Academy of Sciences, Budapest 1364, P. O. Box 127, and Dept. of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA (e-mail: furedi@renyi.hu
Research partially supported by Hungarian National Science Foundation Grants OTKA T032452 and T037846 and by NSF grant DMS 0140692.
)
ANDRÁS GYÁRFÁS
Affiliation:
Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P. O. Box 63, Hungary-1518 (e-mail: gyarfas@sztaki.hu)
GÁBOR SIMONYI
Affiliation:
Alfréd Rényi Institute, Hungarian Academy of Sciences, Budapest 1364, P. O. Box 127 (e-mail: simonyi@renyi.hu
Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846 and T046376.
)
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Extract

Hadwiger's well known conjecture (see the survey of Toft [9]) states that any graph $G$ has a $K_{\chi(G)}$ minor, where $\chi(G)$ is the chromatic number of $G$. Let $\alpha(G)$ denote the independence (or stability) number of $G$, namely the maximum number of pairwise nonadjacent vertices in $G$. It was observed in [1], [4], [10] that via the inequality $\chi(G)\ge {|V(G)|\over \alpha(G)}$, Hadwiger's conjecture implies

Conjecture 1.1.Any graph G on n vertices contains a$K_{\lceil {n\over \alpha(G)}\rceil}$as a minor.

Type
PROBLEM SECTION
Information
Combinatorics, Probability and Computing , Volume 14 , Issue 3 , May 2005 , pp. 435 - 438
Copyright
© 2005 Cambridge University Press

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