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Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
The graph K 1,3 is called a claw, and is used to define the claw-free graphs. [5] The graph K 3,3 is called the utility graph. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the nonplanarity of K 3,3. [6]
Among all such graphs, it is the smallest. Therefore, it is the (3,4)-cage, the smallest graph that has three neighbors per vertex and in which the shortest cycle has length four. [25] Like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two ...
K 5 and K 3,3: Homeomorphic subgraph Kuratowski's theorem: K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph minor Auer et al. (2013) [2] Graphs of fixed genus: A finite obstruction set Graph minor Diestel (2000), [1] p. 275 ...
The form of YΔ- and ΔY-transformations used to define the Petersen family is as follows: . If a graph G contains a vertex v with exactly three neighbors, then the YΔ-transform of G at v is the graph formed by removing v from G and adding edges between each pair of its three neighbors.
All non-isomorphic graphs on 3 vertices and their chromatic polynomials, clockwise from the top. The independent 3-set: k 3. An edge and a single vertex: k 2 (k – 1). The 3-path: k(k – 1) 2. The 3-clique: k(k – 1)(k – 2). The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics.
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...