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More generally, every planar graph of minimum degree at least three either has an edge of total degree a ...{2,n-2}</math> have edges with unbounded total degree. However, for planar graphs with vertices of degree lower than three, variants of the theorem have been ...

{{Short description|On Hamiltonian cycles in planar graphs}} ...Hamiltonian. The result has been widely used to prove that certain planar graphs constructed to have additional properties are not Hamiltonian; for instance ...

{{Short description|Every triangle-free planar graph is 3-colorable}} [[File:Groetsch-3-coloring.svg|thumb|A 3-coloring of a triangle-free planar graph]] ...

...rtex-connected graph|5-vertex-connected]] graph that is not [[planar graph|planar]] contains a [[Homeomorphism (graph theory)|subdivision]] of the 5-vertex [ .... In the other direction, by [[Kuratowski's theorem]], a graph that is not planar necessarily contains a subdivision of either {{math|''K''₅}} or ...

The first half of the book provides the statements of nearly 100 propositions in the discrete geometry of the [[Euclidean plan ...ull)|Carathéodory's theorem]] that every point in the [[convex hull]] of a planar set belongs to a triangle determined by three points of the set, and Steini ...

{{Short description|Any planar graph can be subdivided by removing a few vertices}} ...f [[isoperimetric inequality]] for [[planar graph]]s, that states that any planar graph can be split into smaller pieces by removing a small number of [[Vert ...

...whether or not an [[undirected graph]] has a particular property such as [[planar graph|planarity]] or [[bipartite graph|bipartiteness]]. They are named afte ...es without finding any edges, then report that the graph is empty. On some graphs (for instance the [[complete graph]]s) this algorithm will terminate quickl ...

{{about|the theorem on graphs of polyhedra}} ...the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as [[polyhedral graph]]s.{{r|mathworld}} ...

...than [[Alfred Tarski]]'s [[quantifier elimination]] procedure for deciding statements in the [[Real closed field#Decidability and quantifier elimination|first-or ...are properly quantified, and (when interpreted as a mathematical statement about the [[real number]]s) it is a true statement. As Tarski showed, this theory ...

.../math> onto the edge <math>(u,v)</math> cancel out. Each of the following statements is sufficient for a <math>d</math>-dimensional framework to be infinitesima ...=Asimow|first1=L|last2=Roth|first2=B|date=1979-03-01|title=The rigidity of graphs, II|journal=Journal of Mathematical Analysis and Applications|language=en|v ...