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2. Grötzsch's theorem - Wikipedia

   en.wikipedia.org/wiki/Grötzsch's_theorem

   In 2003, Carsten Thomassen [3] derived an alternative proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free planar graphs that are not 3-list-colorable. [4]

3. Outerplanar graph - Wikipedia

   en.wikipedia.org/wiki/Outerplanar_graph

   In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K 4 and K 2,3, or by their Colin de Verdière graph invariants. They have Hamiltonian ...

4. k-outerplanar graph - Wikipedia

   en.wikipedia.org/wiki/K-Outerplanar_graph

   Baker's technique covers a planar graph with a constant number of -outerplanar graphs and uses their low treewidth in order to quickly approximate several hard graph optimization problems. [ 2 ] In connection with the GNRS conjecture on metric embedding of minor-closed graph families, the k {\displaystyle k} -outerplanar graphs are one of the ...

5. Apollonian network - Wikipedia

   en.wikipedia.org/wiki/Apollonian_network

   The complete graphs on three and four vertices, K 3 and K 4, are both Apollonian networks. K 3 is formed by starting with a triangle and not performing any subdivisions, while K 4 is formed by making a single subdivision before stopping. The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar ...

6. Complete bipartite graph - Wikipedia

   en.wikipedia.org/wiki/Complete_bipartite_graph

   A complete bipartite graph of K 4,7 showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots) For any k, K 1,k is called a star. [2] All complete bipartite graphs which are trees are stars. The graph K 1,3 is called a claw, and is used to define the claw-free graphs ...

7. Complete graph - Wikipedia

   en.wikipedia.org/wiki/Complete_graph

   However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite graph K 3,3 as a subdivision, and by ...

8. Thickness (graph theory) - Wikipedia

   en.wikipedia.org/wiki/Thickness_(graph_theory)

   That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k. [1] [2] In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G. [3] Thus, a planar graph has thickness one.

9. Robertson–Seymour theorem - Wikipedia

   en.wikipedia.org/wiki/Robertson–Seymour_theorem

   Wagner's theorem states that a graph is planar if and only if it has neither K 5 nor K 3,3 as a minor. In other words, the set {K 5, K 3,3} is an obstruction set for the set of all planar graphs, and in fact the unique minimal obstruction set. A similar theorem states that K 4 and K 2,3 are the forbidden minors for the set of outerplanar graphs.