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Double covers of graphs

Published online by Cambridge University Press:  17 April 2009

Derek A. Waller
Derek A. Waller
Affiliation:
Department of Pure Mathematics, University College, Swansea, Wales.
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Abstract

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A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.

In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 2 , April 1976 , pp. 233 - 248
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Biggs, Norman, Algebraic graph theory (Cambridge Tracts in Mathematics, 67. Cambridge University Press, Cambridge, 1974).Google Scholar
[2]Djoković, D.Ž., “Automorphisms of graphs and coverings”, J. Combinatorial Theory Ser. B 16 (1974), 243–247.Google Scholar
[3]Farzan, M. and Waller, D.A., “Kronecker products and local joins of graphs”, submitted.Google Scholar
[4]Gardiner, A., “Antipodal covering graphs”, J. Combinatorial Theory Ser. B 16 (1974), 255–273.Google Scholar
[5]Harary, Frank, Graph theory (Addison-Wesley, Reading, Massachusetts; London; Ontario; 1969).Google Scholar
[6]Husemoller, Dale, Fibre bundles (McGraw-Hill, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1966).Google Scholar
[7]Massey, William S., Algebraic topology: an introduction (Harcourt, Brace and World, New York, 1967).Google Scholar
[8]Smith, D.H., “Primitive and imprimitive graphs”, Quart. J. Math. Oxford Ser. 22 (1971), 551–557.Google Scholar