Journal
JOURNAL OF GRAPH THEORY
Volume 79, Issue 4, Pages 318-330Publisher
WILEY-BLACKWELL
DOI: 10.1002/jgt.21834
Keywords
local chromatic number; oriented graphs; fractional colorings
Categories
Funding
- Hungarian Foundation for Scientific Research Grant [OTKA K76088, K104343, K105840, NK78439, TAMOP - 4.2.2.B-10/1-2010-0009, OTKA NN-102029]
- Lendulet project of the Hungarian Academy of Sciences
- Hungarian Foundation for Scientific Research Grant by the National Office for Research and Technology [TAMOP - 4.2.2.B-10/1-2010-0009, NKTH-OTKA K67651]
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The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.
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