Abstract
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 .
Similar content being viewed by others
References
J. Akiyama, G. Exco andF. Harary, Covering and packing in graphs III. Cyclic and acyclic invariants,Math. Slovaca, to appear.
J. Bosák, Chromatic index of finite and infinite graphs,Czechoslovak Math. J. 22 (1972), 272–290.
D. P. Geller andA. J. W. Hilton, How to colour the lines of a bigraph,Networks 4 (1974), 281–282.
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.
D. König,Theorie der endlichen und unendlichen Graphen, Chelsea Pub. Co. New York (1950).
R. Rado, Axiomatic treatment of rank in infinite sets,Can. J. of Math. 1 (1949), 337–343.
R. Rado, A selection lemma,J. Combinatorial Theory,10 (1971), 176–177.
P. Tomasta, Note on linear arboricity,Math. Slovaca, to appear.
V. G. Vizing, On an estimate of the chromatic class of ap-graph,Diskret. Analiz 3 (1964), 25–30.
V. G. Vizing, Critical graphs with a given chromatic class,Diskret. Analiz 5 (1965), 9–17.
D. de Werra, A few remarks on chromatic scheduling,Combinatorial Programming: Methods and Applications (B. Roy (ed.)), D. Reidel Pub. Co. (1975), 337–342.
D. de Werra, Partial compactness in chromatic scheduling,Proc. III Symposium on Operations Research (Heidelberg, Sept. 1978), Op. Res. Verfahren,32 (1979), 309–316.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hilton, A.J.W. Canonical edge-colourings of locally finite graphs. Combinatorica 2, 37–51 (1982). https://doi.org/10.1007/BF02579280
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02579280