Pentagon-Generated Trivalent Graphs with Girth 5

Abstract

The terminology of [1] will be assumed in what follows. LetPb(G) stand for the set of pentagons in the graphG.Call a graphpentagongeneratedwhen it is the union of its contained pentagons. LetP5,3be 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 thatH∈P5,3. LetEk(H) be the set of edges in exactlyk∈ 0 pentagons ofH. ClearlyEk(H) = 0 ifk≠ 1, 2, 3, 4 and |E1(H) ∩E(P)r ≦ 2, for allP∈P5(H).P∈P5(H) issingularwhen |E1(H) ∩E(P)r = 2,.