Canonical edge-colourings of locally finite graphs


A variety of results on edge-colourings are proved, the main one being the following: ifG is a graph without loops or multiple edges and with maximum degree Δ=Δ(G), and if ν is a given integer 1≦ν≦Δ(G), thenG can be given a proper edge-colouring with the coloursc 1, ...,c Δ+1 with the additional property that any edge colouredc μ with μ≧ν is on a vertex which has on it edges coloured with at least ν − 1 ofc 1, ...,c v .

This is a preview of subscription content, access via your institution.


  1. [1]

    J. Akiyama, G. Exco andF. Harary, Covering and packing in graphs III. Cyclic and acyclic invariants,Math. Slovaca, to appear.

  2. [2]

    J. Bosák, Chromatic index of finite and infinite graphs,Czechoslovak Math. J. 22 (1972), 272–290.

    MathSciNet  Google Scholar 

  3. [3]

    D. P. Geller andA. J. W. Hilton, How to colour the lines of a bigraph,Networks 4 (1974), 281–282.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    A. J. W. Hilton, Colouring the edges of a multigraph so that each vertex has at mostj, or at leastj, edges of each colour on it,J. London Math. Soc. (2),12 (1975), 123–128.

    MATH  Article  Google Scholar 

  5. [5]

    D. König,Theorie der endlichen und unendlichen Graphen, Chelsea Pub. Co. New York (1950).

    Google Scholar 

  6. [6]

    R. Rado, Axiomatic treatment of rank in infinite sets,Can. J. of Math. 1 (1949), 337–343.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    R. Rado, A selection lemma,J. Combinatorial Theory,10 (1971), 176–177.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    P. Tomasta, Note on linear arboricity,Math. Slovaca, to appear.

  9. [9]

    V. G. Vizing, On an estimate of the chromatic class of ap-graph,Diskret. Analiz 3 (1964), 25–30.

    MathSciNet  Google Scholar 

  10. [10]

    V. G. Vizing, Critical graphs with a given chromatic class,Diskret. Analiz 5 (1965), 9–17.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    D. de Werra, A few remarks on chromatic scheduling,Combinatorial Programming: Methods and Applications (B. Roy (ed.)), D. Reidel Pub. Co. (1975), 337–342.

  12. [12]

    D. de Werra, Partial compactness in chromatic scheduling,Proc. III Symposium on Operations Research (Heidelberg, Sept. 1978), Op. Res. Verfahren,32 (1979), 309–316.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hilton, A.J.W. Canonical edge-colourings of locally finite graphs. Combinatorica 2, 37–51 (1982).

Download citation

AMS subject classification (1980)

  • 05 C 15