A note on generalized chromatic number and generalized girth

13 years 4 months ago

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Erdos proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers. Generalized graph coloring describes the partitioning of the vertices into classes whose induced subgraphs satisfy particular constraints. When P is a family of graphs, the P chromatic number of a graph G, written P, is the minimum size of a partition of V (G) into classes that induce subgraphs of G belonging to P. When P is the family of independent sets, P is the ordinary chromatic number. General aspects are studied in [1-3,7-9,11-14,1718]. Many additional results are known about particular generalized chromatic numbers. One aim in the study of generalized chromatic numbers is the extension of classical coloring results. Erdos [4] proved that there exist graphs of large chromatic number and large girth. We study the extension of this for a class of generalized coloring parameters. We consider the family P consisting of all graphs not...

Béla Bollobás, Douglas B. West

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