Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
Hamiltonian decomposition of Cayley graphs of degree 4
Journal of Combinatorial Theory Series B
Bounding the number of embeddings of 5-connected projective-planar graphs
Journal of Graph Theory
Arrangement graphs: a class of generalized star graphs
Information Processing Letters
4-connected projective-planar graphs are Hamiltonian
Journal of Combinatorial Theory Series B
Finding Hamiltonian circuits in arrangements of Jordan curves is NP-complete
Information Processing Letters
Basic graph theory: paths and circuits
Handbook of combinatorics (vol. 1)
Extremal problems in combinatorial geometry
Handbook of combinatorics (vol. 1)
Handbook of discrete and computational geometry
Sweeps, arrangements and signotopes
Discrete Applied Mathematics - Special issue 14th European workshop on computational geometry CG'98 Selected papers
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We study connectivity, Hamilton path and Hamilton cycle decomposition, 4-edge and 3-vertex coloring for geometric graphs arising from pseudoline (affine or projective) and pseudocircle (spherical) arrangements. While arrangements as geometric objects are well studied in discrete and computational geometry, their graph theoretical properties seem to have received little attention so far. In this paper we show that they provide well-structured examples of families of planar and projective-planar graphs with very interesting properties. Most prominently, spherical arrangements admit decompositions into two Hamilton cycles; this is a new addition to the relatively few families of 4-regular graphs that are known to have Hamiltonian decompositions. Other classes of arrangements have interesting properties as well: 4-connectivity, 3-vertex coloring or Hamilton paths and cycles. We show a number of negative results as well: there are projective arrangements which cannot be 3-vertex colored. A number of conjectures and open questions accompany our results.