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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 .
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 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 ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
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 ...
Where a coloring of a graph is a mapping from the vertex set to the natural numbers, a coloring of a signed graph is a mapping from the vertex set to the integers. The constraints on proper colorings come from the edges of the signed graph. The integers assigned to two vertices must be distinct if they are connected by a positive edge.
A partition of the edges of the complete graph K 5 into two triangle-free subsets. In graph theory and theoretical computer science, the monochromatic triangle problem is an algorithmic problem on graphs, in which the goal is to partition the edges of a given graph into two triangle-free subgraphs. It is NP-complete but fixed-parameter ...
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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