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Long cycles and spanning subgraphs of locally maximal 1-planar graphs. (English) Zbl 1486.05072

Summary: A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph \(G \), we show the existence of a spanning 3-connected planar subgraph and prove that \(G\) is Hamiltonian if \(G\) has at most three 3-vertex-cuts, and that \(G\) is traceable if \(G\) has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.

MSC:

05C12 Distance in graphs
05C38 Paths and cycles
05C10 Planar graphs; geometric and topological aspects of graph theory
05C45 Eulerian and Hamiltonian graphs
05C62 Graph representations (geometric and intersection representations, etc.)
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