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Pentagon-Generated Trivalent Graphs with Girth 5

Published online by Cambridge University Press:  20 November 2018

Neil Robertson
Neil Robertson*
Affiliation:
The Ohio State University, Columbus, Ohio
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The terminology of [1] will be assumed in what follows. Let Pb(G) stand for the set of pentagons in the graph G. Call a graph pentagongenerated when it is the union of its contained pentagons. Let P5,3 be the class of connected trivalent pentagon-generated graphs with girth 5. These graphs form a family including the Petersen graph and the graph of the dodecahedron. They are studied here and completely classified in terms of a decomposition which all but some specifically determined indecomposable graphs admit.

Assume henceforth that H ∈ P5,3. Let Ek(H) be the set of edges in exactly k ∈ 0 pentagons of H. Clearly Ek(H) = 0 if k ≠ 1, 2, 3, 4 and |E1(H) ∩ E(P)r ≦ 2, for all P ∈ P5(H). P ∈ P5(H) is singular when |E1(H) ∩ E(P)r = 2,.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 36 - 47
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Tutte, W. T., Connectivity in graphs, Mathematical Expositions, No. 15 (Univ. Toronto Press, Toronto, Ontario; Oxford Univ. Press, London, 1966).Google Scholar