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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 ...
The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors. Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored. [8]
A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, [5] a directed graph is an ordered pair = (,) comprising: , a set of vertices (also called nodes or points);
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 ...
In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. [ 1 ] The triangle graph is also known as the cycle graph C 3 {\displaystyle C_{3}} and the complete graph K 3 {\displaystyle K_{3}} .
More precisely one speaks of Shannon multigraph Sh(n), if the three vertices are connected by ⌊ ⌋, ⌊ ⌋ and ⌊ + ⌋ edges respectively. This multigraph has maximum degree n . Its multiplicity (the maximum number of edges in a set of edges that all have the same endpoints) is ⌊ n + 1 2 ⌋ {\displaystyle \left\lfloor {\frac {n+1}{2 ...
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