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Pentagon-Generated Trivalent Graphs with Girth 5
Published online by Cambridge University Press: 20 November 2018
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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,.
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- Copyright © Canadian Mathematical Society 1971
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