Graphs and Combinatorics

, Volume 17, Issue 4, pp 681–685

A Generalization of the Gallai–Roy Theorem

DOI: 10.1007/PL00007256

Cite this article as:
Li, H. Graphs Comb (2001) 17: 681. doi:10.1007/PL00007256

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.

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