Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T13:51:58.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Star multigraphs with three vertices of maximum degree

Published online by Cambridge University Press:  24 October 2008

A. G. Chetwynd  and
A. J. W. Hilton
A. G. Chetwynd
Affiliation:
Department of Mathematics, University of Lancaster
A. J. W. Hilton
Affiliation:
Department of Mathematics, University of Reading
Get access Rights & Permissions [Opens in a new window]

Extract

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,

where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 2 , September 1986 , pp. 303 - 317
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Beineke, L. W. and Fiorini, S.. On small graphs critical with respect to edge-colourings. Discrete Math. 16 (1976), 15–20.CrossRefGoogle Scholar
[2] Berge, C.. Graphs and Hypergraphs (North-Holland, 1973).Google Scholar
[3] Chetwynd, A. G. and Hilton, A. J. W.. Partial edge-colourings of complete graphs or of graphs which are nearly complete. Graph Theory and Combinatorics (Academic Press, vol. in honour of P. Erdös' 70th birthday), 1984, 81–98.Google Scholar
[4] Chetwynd, A. G. and Hilton, A. J. W.. The chromatic index of graphs of even order with many edges. J. Graph Theory 8 (1984), 463–470.CrossRefGoogle Scholar
[5] Chetwynd, A. G. and Hilton, A. J. W.. Regular graphs of high degree are 1-factorizable. Proc. London Math. Soc. 50 (1985), 193–206.CrossRefGoogle Scholar
[6] Chetwynd, A. G. and Hilton, A. J. W.. Critical star multigraphs. Submitted.Google Scholar
[7] Chetwynd, A. G. and Hilton, A. J. W.. The edge-chromatic class of graphs with maximum degree at least |V| – 3. Submitted.Google Scholar
[8] Chetwynd, A. G. and Hilton, A. J. W.. The edge-chromatic class of graphs of even order with maximum degree at least |V| – 4. (In preparation.)Google Scholar
[9] Chetwynd, A. G. and Yap, H. P.. Chromatic index critical graphs of order 9. Discrete Math. 47 (1983), 23–33.CrossRefGoogle Scholar
[10] Chvátal, V.. On Hamilton´s ideals. J. Comb. Theory (B), 12 (1972), 163–168.CrossRefGoogle Scholar
[11] Hilton, A. J. W. and Rodger, C. A.. Triangulating nearly complete graphs of odd order. (In preparation.)Google Scholar
[12] Jakobsen, I. T.. On critical graphs with chromatic index 4. Discrete Math. 9 (1974), 265–276.CrossRefGoogle Scholar
[13] Plantholt, M.. The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981), 5–13.CrossRefGoogle Scholar
[14] Vizing, V. G.. On an estimate of the chromatic class of a p-graph [in Russian]. Diskret. Analiz. 3 (1964), 25–30.Google Scholar