Generating locally-cyclic triangulations of surfaces
type: Item
instance of: scholarly article
title: Generating locally-cyclic triangulations of surfaces
zbMATH Open document ID: 0723.05053
DOI: 10.1016/0095-8956(92)90015-P
author: Aleksander Malnič
author: Bojan Mohar
published in: Journal of Combinatorial Theory. Series B
publication date: 1992-01-01T00:00:00Z
review text: A 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.
reviewed by: Bojan Mohar
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 4191682
zbMATH Keywords: locally-cyclic graph
zbMATH Keywords: surfaces
zbMATH Keywords: triangulations
zbMATH Keywords: vertex-splitting
zbMATH Keywords: double cycle cover
Wikidata QID: Q56504290
MaRDI profile type: Publication
cites work: Densely embedded graphs
cites work: Curves von 2-manifolds and isotopies
cites work: Q3757929
cites work: Clean triangulations
cites work: Q5534009
cites work: Q3972103
cites work: Q3972954
cites work: Embeddings of graphs with no short noncontractible cycles
cites work: Q4273853
DBLP publication ID: journals/jct/MalnicM92
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description: scientific article; zbMATH DE number 4191682