Local Chromatic Number, KY Fan's Theorem, And Circular Colorings |
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Authors: | Gábor Simonyi Gábor Tardos? |
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Institution: | (1) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary;(2) School of Computing Science, Simon Fraser University, Burnaby BC, V5A 1S6, Canada;(3) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary |
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Abstract: | The local chromatic number of a graph was introduced in 14]. It is in between the chromatic and fractional chromatic numbers.
This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include
Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their
generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.
We use an old topological result of Ky Fan 17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored
copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan 18].) This yields a
lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.
As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers
if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by
F. Meunier 42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can
be arbitrarily close to the other extreme.
* Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826,
and NK62321.
† Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA)
Nos. T037846, T046234, AT048826, and NK62321. |
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Keywords: | 05C15 55U10 |
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