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Excluding Minors in Cubic Graphs

Published online by Cambridge University Press:  12 September 2008

K. Kilakos  and
B. Shepherd
K. Kilakos
Affiliation:
Centre for Discrete and Applicable Mathematics, London School of Economics, London, UK
B. Shepherd
Affiliation:
Centre for Discrete and Applicable Mathematics, London School of Economics, London, UK
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Abstract

Let P10\e be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P10\e. The decomposition is used to show that graphs in this class are 3-edge-colourable. We also consider an application to a conjecture due to Grötzsch which states that a planar graph is 3-edge-colourable if and only if it is fractionally 3-edge-colourable.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 1 , March 1996 , pp. 57 - 78
Copyright
Copyright © Cambridge University Press 1996

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References

[1]Alspach, B., Goddyn, L. A. and Zhang, C. Q. (1996) Graphs with the circuit cover property. Submitted.Google Scholar
[2]Appel, K. and Haken, W. (1977) Every planar map is four-colourable, part I: discharging, Illinois J. Math. 21 429–490.Google Scholar
[3]Appel, K., Haken, W. and Koch, J. (1977) Every planar map is four-colourable, part II: reducibility. Illinois J. Math. 21 491–567.Google Scholar
[4]Edmonds, J. (1965) Maximum matching and a polyhedron with (0,1) vertices. J. Res. Nat. Bur. Standards Sect. B, 69B 125–130.Google Scholar
[5]Ellingham, M. N. (1984) Petersen subdivisions in some regular graphs. Congressus Numerantium 44 33–40.Google Scholar
[6]Fontet, M. (1978) Graphes 4-essentiels. C.R. Acad. Sci. Paris t287 289–290.Google Scholar
[7]Goddyn, L. A. (1996) Cones, lattices and Hilbert bases of circuits and perfect matchings. Contemporary Mathematics (to appear).Google Scholar
[8]Holyer, I. (1981) The NP-completeness of edge-colouring. SIAM J. Computing 10 718–720.CrossRefGoogle Scholar
[9]Kelmans, A. K. (1986) On 3-connected graphs without essential 3-cuts or triangles. Soviet Math. Dokl. 33(3) 698–703.Google Scholar
[10]Lovász, L. (1987) Matching structure and the matching lattice. J. Combinatorial Theory (B) 43 187–222.CrossRefGoogle Scholar
[11]Seymour, P. D. (1979) On multi-colourings of cubic graphs and conjectures of Fulkerson and Tutte. Proc. London Math. Society (3) 38 423–460.CrossRefGoogle Scholar
[12]Seymour, P. D. (1981) On Tutte's extension of the four-colour problem. J. Combinatorial Theory (B) 31 82–94.CrossRefGoogle Scholar
[13]Tait, P. G. (1880) Remarks on the colouring of maps. Proc. Royal Soc. Edinburgh, Ser. A 10 729.Google Scholar
[14]Tutte, W. T. (1966) On the algebraic theory of graph colourings. J. Combinatorial Theory (B) 23 15–50.CrossRefGoogle Scholar
[15]Wagner, K. (1937) Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114 570–590.CrossRefGoogle Scholar
[16]Wormald, N. C. (1979) Classifying k-connected cubic graphs. Lecture Notes in Mathematics 748. Springer Verlag, 199–206.Google Scholar