In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect...

In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing...

no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that share...

mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each...

terms of graph theory, by considering it in terms of constructing a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic...

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...

path of two adjacent edges. In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with maximum degree six or seven are...

special case of 3-connected cubic planar graphs (Heckman & Krakovski 2013) Weaker results relating the degree of a graph to unavoidable sets of cycle lengths...

In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the...

the graph is planar, then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a graph embedding...

a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that K 3 , 3 {\displaystyle K_{3,3}} is not a planar graph. Multiple...

either its maximum degree Δ or Δ+1. For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs...

vertices in the whole graph. Theorem—A 4-connected planar triangulation has a Hamiltonian cycle. Theorem—A 4-connected planar graph has a Hamiltonian cycle...

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split...

The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete...

graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5...

at these crossings. In graph theory terms, a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated...

In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can...

In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists...

every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than...

is planar, together imply that the planar graphs are (3,6)-sparse. However, not every (3,6)-sparse graph is planar. Similarly, outerplanar graphs are...

In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer...

apex graph is a graph in which one vertex can be removed, leaving a planar subgraph. The removed vertex is called the apex. A k-apex graph is a graph that...

vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can...

bipartite graph, testing whether it contains a complete bipartite subgraph Ki,i for a parameter i is an NP-complete problem. A planar graph cannot contain...

the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, and that bounded-degree planar graphs...

proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian simplicial...

if a topological graph is 2-quasi-planar, then it is a planar graph. It follows from Euler's polyhedral formula that every planar graph with n > 2 vertices...

In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states...

In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who...