Linked from

The 37 pages that link to Planar graph, each with the reason it gives.

• Graph theoryBroader topic: Planarity links graph structure to geometric embedding and coloring constraints.

• Bipartite graphCompared with: Planarity and bipartiteness are distinct properties, though together they constrain possible cycles.

• Graph coloringRelated: Planarity imposes a famous upper bound of four on the chromatic number.

• Vertex (geometry)Related: In a planar drawing, vertices mark the endpoints and meeting points of edges.

• Four color theoremNarrower topic: Map regions become vertices, and shared boundary segments become edges.

• Simple graphBroader topic: Planarity is a structural property often studied for simple graphs.

• Euclidean planeRelated: The plane provides the setting for deciding whether a graph admits such a drawing.

• Perfect matchingRelated: Planarity enables special methods for counting perfect matchings.

• Circle packingNarrower topic: Circle tangency patterns correspond to planar graphs.

• Graph minorBroader topic: Planarity is characterized by excluding two specific graph minors.

• DiagonalRelated: Polygon diagonals create additional edges while remaining within the polygon's planar drawing.

• Undirected graphRelated: Planarity is a structural constraint frequently studied for undirected networks.

• Euler's polyhedron formulaRelated: Projecting a polyhedron's edge network onto a plane turns its count into a graph relation.

• Graph embeddingBroader topic: Planarity asks whether a graph embeds in the plane without crossings.

• Hamiltonian cycleRelated: Hamiltonian-cycle existence remains NP-complete even when restricted to planar graphs.

• Handshaking lemmaRelated: Degree counting combines with face counts to establish planar graph bounds.

• Petersen graphCompared with: The Petersen graph cannot be drawn without crossings, despite its small size.

• Chromatic polynomialBroader topic: The four-color theorem asserts every planar graph has a proper coloring with at most four colors.

• Planar separator theoremBroader topic: Planarity is the structural assumption that makes the separator bound possible.

• Robertson–Seymour theoremRelated: Planarity exemplifies a property captured by finitely many forbidden minors.

• Wagner's theoremNarrower topic: Wagner’s theorem identifies exactly which finite graphs belong to this class.

• Circle packing theoremNarrower topic: The theorem translates this graph structure into circle tangencies.

• CoplanarityRelated: Its drawings depend on placing vertices and edges consistently within one plane.

• Edge contractionRelated: Contracting an edge in a planar graph preserves planarity.

• Venn diagramRelated: Curve intersections and the regions they enclose are constrained by planar geometry.

• Fáry's theoremNarrower topic: Fáry's theorem begins with this topological condition and guarantees a straight-line drawing.

• Five color theoremNarrower topic: The theorem applies precisely to this class of graphs.

• Intersection graphCompared with: Planarity constrains drawings of edges, unlike intersection graphs, which constrain represented objects.

• Burr–Erdős conjectureBroader topic: Every planar graph is 5-degenerate, so the conjecture predicts linear Ramsey growth for this family.

• Conway's thrackle conjectureCompared with: Thrackle drawings permit crossings, unlike planar embeddings.

• Erdős–Pósa theoremCompared with: Cycles in planar graphs have stronger linear packing-covering bounds than arbitrary graphs.

• FKT algorithmNarrower topic: Planarity enables the signing conditions that make the Pfaffian count exact.

• Graph structure theoremRelated: For excluded minors that force planarity, the structural pieces are close to planar.

• Grötzsch's theoremNarrower topic: Planarity is the structural condition that makes the three-color bound possible.

• Honeycomb theoremRelated: Cell boundaries form a planar graph, linking local edge counts to global constraints.

• Three utilities problemNarrower topic: The puzzle asks whether K₃,₃ is planar; it is not.

• Two-dimensional spaceBroader topic: Planar graphs encode relationships constrained by a two-dimensional embedding.