, Volume 37, Issue 4, pp 785–793 | Cite as

Infinitely connected subgraphs in graphs of uncountable chromatic number

  • Carsten Thomassen
Original paper


Erdős and Hajnal conjectured in 1966 that every graph of uncountable chromatic number contains a subgraph of infinite connectivity. We prove that every graph of uncountable chromatic number has a subgraph which has uncountable chromatic number and infinite edge-connectivity. We also prove that, if each orientation of a graph G has a vertex of infinite outdegree, then G contains an uncountable subgraph of infinite edge-connectivity.

Mathematics Subject Classification (2000)

05C15 05C40 05C63 


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  1. [1]
    P. Erdős and A. Hajnal: On chromatic number of graphs and set systems, Acta Math. Acad. Sci. Hungar 17 (1966), 61–99.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    G. Fodor: Proof of a conjecture of P. Erdős, Acta Sci. Math (Szeged). 14 (1951), 219–227.zbMATHGoogle Scholar
  3. [3]
    A. Hajnal and P. Komjáth: What must and what need not be contained in a graph of uncountable chromatic number, Combinatorica 4 (1984), 47–52.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. L. Hakimi: On the degree of the vertices of a directed graph, J. Franklin Institute 279 (1965), 290–308.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Komjáth: Connectivity and chromatic number of infinite graphs, Israel J. Math 56 (1986), 257–266.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Komjáth: The chromatic number of infinite graphs-a survey, Discrete Math 311 (2011), 1448–1450.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    P. Komjáth: A note on chromatic number and connectivity of infinite graphs, Israel J. Math 196 (2013), 499–506.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P. Komjáth: The list-chromatic number of infinite graphs, manuscript.Google Scholar
  9. [9]
    D. T. Soukup: Trees, ladders, and graphs, J. Combin. Theory Ser. B 115 (2015), 96–116.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C. Thomassen: Infinite graphs, in: Further Selected Topics in Graph Theory (L. W. Beineke and R. J. Wilson, eds.) Academic Press, London (1983) 129–160.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

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