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Graphs and Combinatorics

, Volume 17, Issue 4, pp 681–685 | Cite as

A Generalization of the Gallai–Roy Theorem

Abstract.

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex vV(G), there is a path P starting at v which represents all χ(G) colors.

Keywords

Directed Path Positive Answer Chromatic Number Essential Result Proper Coloring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Tokyo 2001

Authors and Affiliations

  • Hao Li
    • 1
  1. 1.Laboratoire de Recherche en Informatique, URA 410, C.N.R.S., Bât. 490, Université de Paris-sud, 91405-Orsay cedex, France. e-mail: li@lri.frFR

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