On topological relaxations of chromatic conjectures

Authors:
Gábor Simonyi;Ambrus Zsbán
Affiliations:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Hungary;Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in Simonyi and Tardos (2006) [41], our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.