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A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints. A bipartite graph (,,) may be used to model a hypergraph in which U is the set of vertices of the hypergraph, V is the set of hyperedges, and ...
In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 . The edges of an undirected simple graph permitting loops induce a symmetric homogeneous relation on the vertices of that is called the adjacency relation of .
A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph K n is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph ...
Claw-free graphs are the graphs that are locally co-triangle-free; that is, for all vertices, the complement graph of the neighbourhood of the vertex does not contain a triangle. A graph that is locally H is claw-free if and only if the independence number of H is at most two; for instance, the graph of the regular icosahedron is claw-free ...
Here, it is emphasized that only degree-2 (i.e., 2-valent) vertices can be smoothed. The limit of this operation is realized by the graph that has no more degree-2 vertices. For example, the simple connected graph with two edges, e 1 {u,w } and e 2 {w,v }: has a vertex (namely w) that can be smoothed away, resulting in:
Each small triangle becomes a node in the new graph derived from the triangulation. The small letters identify the areas, eight inside the figure, and area i designates the space outside of it. As described previously, those nodes that share an edge whose endpoints are numbered 1 and 2 are joined in the derived graph.
This nine-edge Shannon multigraph requires nine colors in any edge coloring; its vertex degree is six and its multiplicity is three. According to a theorem of Shannon (1949) , every multigraph with maximum degree Δ {\displaystyle \Delta } has an edge coloring that uses at most 3 2 Δ {\displaystyle {\frac {3}{2}}\Delta } colors.
Whenever two edges of a Euclidean minimum spanning tree meet at a vertex, they must form an angle of 60° or more, with equality only when they form two sides of an equilateral triangle. This is because, for two edges forming any sharper angle, one of the two edges could be replaced by the third, shorter edge of the triangle they form, forming ...
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