Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T19:14:05.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Graphs which are vertex-critical with respect to the edge-chromatic number

Published online by Cambridge University Press:  24 October 2008

A. J. W. Hilton  and
P. D. Johnson
A. J. W. Hilton
Affiliation:
Department of Mathematics, University of Reading, Reading RG6 2AX
P. D. Johnson
Affiliation:
Department of Algebra, Combinatorics and Analysis, Auburn University, Auburn, AL 36849, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper, multigraphs will have no loops. For a multigraph G, the least number of colours needed to colour the edges of G in such a way that no two edges on the same vertex of G have the same colour, is called the edge-chromatic number, or the chromatic index, of G, and denoted χ′(G). It is clear that if Δ(G) denotes the maximum degree of G, then Δ(G) ≤ χ′(G). If Δ(G) = χ′(G), then G is Class 1, and otherwise G is Class 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 211 - 221
Copyright
Copyright © Cambridge Philosophical Society 1987

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]Chetwynd, A. G. and Hilton, A. J. W.. Partial edge-colourings of graphs or of graphs which are nearly complete. Graph Theory and Combinatorics (Vol. in honour of P. Erdös' 70th birthday), (Academic Press, 1984), 81–98.Google Scholar
[2]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
[3]Chetwynd, A. G. and Hilton, A. J. W.. Regular graphs of high degree are l-factorizable. Proc. London Math. Soc. 50 (1985), 193–206.CrossRefGoogle Scholar
[4]Chetwynd, A. G. and Hilton, A. J. W.. Critical star multigraphs. Graphs and Combinatorics 2 (1986), 209–221.CrossRefGoogle Scholar
[5]Chetwynd, A. G. and Hilton, A. J. W.. The edge-chromatic class of graphs with maximum degree at least |V| –3. Proc. of conf. in memory of G. A. Dirac, to appear.Google Scholar
[6]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
[7]Chetwynd, A. G. and Hilton, A. J. W.. Star multigraphs with three vertices of maximum degree. Math. Proc. Cambridge Philos. Soc. 100 (1986), 303–317.CrossRefGoogle Scholar
[8]Chetwynd, A. G. and Wilson, R. J.. Snarks and supersnarks. The Theory and Applications of Graphs (Wiley, 1981), 215–241.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]Fiorini, S.. Counterexamples to two conjectures of Hilton. J. Graph Theory 2 (1978), 261–264.CrossRefGoogle Scholar
[11]Goldberg, M. K.. Trivalent graphs with chromatic index 4 (in Russian). Soobšč. Acad. Nauk. Gruzin. SSR 93 (1979), 29–31.Google Scholar
[12]Hajós, G.. Über eine Konstruktion nicht n-färberer Graphen. Wiss. Z. Martin-Luther Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 116–117.Google Scholar
[13]Hilton, A. J. W.. Definitions of criticality with respect to edge-colouring. J. Graph Theory 1 (1977), 61–68.CrossRefGoogle Scholar
[14]Hilton, A. J. W. and Johnson, P. D.. Graphs which are vertex-critical with respect to the edge-chromatic class. Submitted to J. Graph Theory.Google Scholar
[15]Jakobsen, I. T.. Some remarks on the chromatic index of a graph. Arch. Math. (Basel) 24 (1973), 440–448.CrossRefGoogle Scholar
[16]Plantholt, M.. The chromatic class of graphs with a spanning star. J. Graph Theory 5 (1981), 5–13.CrossRefGoogle Scholar
[17]Plantholt, M.. On the chromatic index of graphs with large maximum degree. Discrete Math. 47 (1983), 91–96.CrossRefGoogle Scholar
[18]Vizing, V. G.. On an estimate of the chromatic class of p-graph (in Russian). Diskret. Analiz. 3 (1964), 25–30.Google Scholar
[19]Yap, H. P.. On the critical graph conjecture. J. Graph Theory 4 (1980), 309–314.CrossRefGoogle Scholar