Long cycles and spanning subgraphs of locally maximal 1‐planar graphs. Issue 1 (10th February 2020)

• Record Type: Journal Article
• Authors: Fabrici, I.; Harant, J.; Madaras, T.; Mohr, S.; Soták, R.; Zamfirescu, C. T.
• Is Part Of: Journal of graph theory. Volume 95:Issue 1(2020)
• Page Start: 125
• Page End: 137
• Publication Date: 2020-02-10
• Subjects: Hamiltonicity -- longest cycle -- 1‐planar graph -- spanning subgraph; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22542
• Languages: English
• ISSNs: 0364-9024

Abstract

Abstract: 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.