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Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices. [1] Another example of a map graph is the king's graph, a ...
A map M is regular if Aut(M) acts regularly on the flags. Aut(M) of a regular map is transitive on the vertices, edges, and faces of M. A map M is said to be reflexible iff Aut(M) is regular and contains an automorphism that fixes both a vertex v and a face f, but reverses the order of the
The map f is a surjection: each vertex of H has a preimage in C. Furthermore, f maps bijectively each neighbourhood of a vertex v in C onto the neighbourhood of the vertex f(v) in H. For example, let v be one of the purple vertices in C; it has two neighbours in C, a green vertex u and a blue vertex t.
A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
A subdivision of a graph results from inserting vertices into edges (for example, changing an edge • —— • to • — • — • ) zero or more times. An example of a graph with no K 5 or K 3,3 subgraph. However, it contains a subdivision of K 3,3 and is therefore non-planar.
An undirected graph has an Eulerian trail if and only if exactly zero or two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component. [ 6 ] A directed graph has an Eulerian cycle if and only if every vertex has equal in degree and out degree , and all of its vertices with nonzero degree ...
The Grötzsch graph is an example of a 4-chromatic graph without a triangle, and the example can be generalized to the Mycielskians. Theorem (William T. Tutte 1947, [10] Alexander Zykov 1949, Jan Mycielski 1955): There exist triangle-free graphs with arbitrarily high chromatic number.
A map with twelve pentagonal faces. In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the surface and forming connected components (faces) of the complement of the graph.