Long Cycles and Spanning Subgraphs of Locally Maximal 1-planar Graphs

TL;DR

This paper investigates the properties of locally maximal 1-planar graphs, demonstrating the existence of spanning 3-connected planar subgraphs and establishing conditions for Hamiltonicity and traceability based on vertex cuts.

Contribution

It introduces new results on the structure and Hamiltonian properties of 3-connected locally maximal 1-planar graphs, including spanning subgraphs and non-traceable examples.

Findings

Existence of spanning 3-connected planar subgraphs in 3-connected locally maximal 1-planar graphs.

Hamiltonicity is guaranteed if the graph has at most three 3-vertex-cuts.

The paper presents infinitely many non-traceable 5-connected 1-planar graphs.

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 non-traceable $5$-connected $1$-planar graphs are presented.