Abstract
The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.
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The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.
The research of the third author was supported by the Sacta-Rashi Foundation.
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Aharoni, R., Berger, E. & Ziv, R. Independent systems of representatives in weighted graphs. Combinatorica 27, 253–267 (2007). https://doi.org/10.1007/s00493-007-2086-y
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DOI: https://doi.org/10.1007/s00493-007-2086-y