Generating locally-cyclic triangulations of surfaces.

*(English)*Zbl 0723.05053A locally-cyclic graph is a connected graph such that for each vertex the induced subgraph on the set of its neighbors is isomorphic to a cycle. These graphs correspond uniquely to locally-cyclic triangulations of closed surfaces, that is triangulations where each cycle of length three in the underlying graph is facial.

For each closed surface \(\Sigma\), all locally-cyclic triangulations of \(\Sigma\) can be obtained from a minimal basic set B(\(\Sigma\)) by applying the vertex-splitting operation. The basis for the sphere consists of just two graphs, \(K_ 4\) (the tetrahedron) and \(O_ 3\) (the octahedron). Recently, the basis for the projective plane has been found by the second author and coauthors. For all closed surfaces, different from the two sphere, the basis may be characterized by the fact that a locally-cyclic triangulation belongs to the basis if and only if each edge in the underlying graph belongs to some 4-cycle which is homotopically non- trivial. The main result of the paper proves that for an arbitrary closed orientable surface \(\Sigma\), B(\(\Sigma\)) is finite. An application to the study of closed 2-cell embeddings of graphs in surfaces connected to the double cycle cover conjecture is presented.

For each closed surface \(\Sigma\), all locally-cyclic triangulations of \(\Sigma\) can be obtained from a minimal basic set B(\(\Sigma\)) by applying the vertex-splitting operation. The basis for the sphere consists of just two graphs, \(K_ 4\) (the tetrahedron) and \(O_ 3\) (the octahedron). Recently, the basis for the projective plane has been found by the second author and coauthors. For all closed surfaces, different from the two sphere, the basis may be characterized by the fact that a locally-cyclic triangulation belongs to the basis if and only if each edge in the underlying graph belongs to some 4-cycle which is homotopically non- trivial. The main result of the paper proves that for an arbitrary closed orientable surface \(\Sigma\), B(\(\Sigma\)) is finite. An application to the study of closed 2-cell embeddings of graphs in surfaces connected to the double cycle cover conjecture is presented.

Reviewer: B.Mohar (Ljubljana)

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

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\textit{A. Malnič} and \textit{B. Mohar}, J. Comb. Theory, Ser. B 56, No. 2, 147--164 (1992; Zbl 0723.05053)

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