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- Scheinerman's_conjecture abstract "In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976). It was proven by Jeremie Chalopin and Daniel Gonçalves (2009).For instance, the graph G shown below to the left may be represented as the intersection graph of the set of segments shown below to the right. Here, vertices of G are represented by straight line segments and edges of G are represented by intersection points.Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and West (1991) conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directionsand no two segments belong to the same line, then the graph can be colored using k colors, one color for each direction. Therefore, if every planar graph can be represented in this way with only four directions,then the four color theorem follows.Hartman, Newman & Ziv (1991) and de Fraysseix, Ossona de Mendez & Pach (1991) proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also Czyzowicz, Kranakis & Urrutia (1998). De Castro et al. (2002) proved that every triangle-free planar graph can be represented as an intersection graph of line segments having only three directions; this result implies Grötzsch's theorem (Grötzsch 1959) that triangle-free planar graphs can be colored with three colors. de Fraysseix & Ossona de Mendez (2005) proved that if a planar graph G can be 4-colored in such a way that no separating cycle uses all four colors, then G has a representation as an intersection graph of segments.Chalopin, Gonçalves & Ochem (2007) proved that planar graphs are in 1-STRING, the class of intersection graphs of simple curves in the plane that intersect each other in at most one crossing point per pair. This class is intermediate between the intersection graphs of segments appearing in Scheinerman's conjecture and the intersection graphs of unrestricted simple curves from the result of Ehrlich et al. It can also be viewed as a generalization of the circle packing theorem, which shows the same result when curves are allowed to intersect in a tangent. The proof of the conjecture by Chalopin & Gonçalves (2009) was based on an improvement of this result.".
- Scheinerman's_conjecture thumbnail Intersect1.png?width=300.
- Scheinerman's_conjecture wikiPageExternalLink deCastro+2002.6.1.pdf.
- Scheinerman's_conjecture wikiPageExternalLink ChalopinG09.pdf.
- Scheinerman's_conjecture wikiPageID "3125930".
- Scheinerman's_conjecture wikiPageRevisionID "567151498".
- Scheinerman's_conjecture first "Daniel".
- Scheinerman's_conjecture first "Jeremie".
- Scheinerman's_conjecture hasPhotoCollection Scheinerman's_conjecture.
- Scheinerman's_conjecture last "Chalopin".
- Scheinerman's_conjecture last "Gonçalves".
- Scheinerman's_conjecture year "2009".
- Scheinerman's_conjecture subject Category:Conjectures.
- Scheinerman's_conjecture subject Category:Planar_graphs.
- Scheinerman's_conjecture type Abstraction100002137.
- Scheinerman's_conjecture type Cognition100023271.
- Scheinerman's_conjecture type Communication100033020.
- Scheinerman's_conjecture type Concept105835747.
- Scheinerman's_conjecture type Conjectures.
- Scheinerman's_conjecture type Content105809192.
- Scheinerman's_conjecture type Graph107000195.
- Scheinerman's_conjecture type Hypothesis105888929.
- Scheinerman's_conjecture type Idea105833840.
- Scheinerman's_conjecture type PlanarGraphs.
- Scheinerman's_conjecture type PsychologicalFeature100023100.
- Scheinerman's_conjecture type Speculation105891783.
- Scheinerman's_conjecture type VisualCommunication106873252.
- Scheinerman's_conjecture comment "In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976).".
- Scheinerman's_conjecture label "Scheinerman's conjecture".
- Scheinerman's_conjecture sameAs m.08t3g6.
- Scheinerman's_conjecture sameAs Q7431096.
- Scheinerman's_conjecture sameAs Q7431096.
- Scheinerman's_conjecture sameAs Scheinerman's_conjecture.
- Scheinerman's_conjecture wasDerivedFrom Scheinerman's_conjecture?oldid=567151498.
- Scheinerman's_conjecture depiction Intersect1.png.
- Scheinerman's_conjecture isPrimaryTopicOf Scheinerman's_conjecture.
