In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges.
The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph..
By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship graphs are generalized by the triangular cactus graphs.
Friendship theorem The friendship theorem of . states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property..
A combinatorial proof of the friendship theorem was given by Mertzios and Unger. Another proof was given by Craig Huneke. A formalised proof in Metamath was reported by Alexander van der Vekens in October 2018 on the Metamath mailing list.
Labeling and colouring The friendship graph has chromatic number 3 and chromatic index . Its chromatic polynomial can be deduced from the chromatic polynomial of the cycle graph and is equal to .
The friendship graph is edge-graceful if and only if is odd. It is graceful if and only if or ...
Every friendship graph is factor-critical.
Extremal graph theory According to extremal graph theory, every graph with sufficiently many edges (relative to its number of vertices) must contain a -fan as a subgraph. More specifically, this is true for an -vertex graph (for sufficiently large in terms of ) if the number of edges is
where is if is odd, and is if is even. These bounds generalize Turán's theorem on the number of edges in a triangle-free graph, and they are the best possible bounds for this problem (when ), in that for any smaller number of edges there exist graphs that do not contain a -fan..
Generalizations
Any two vertices having exactly one neighbor in common is equivalent to any two vertices being connected by exactly one path of length two. This has been generalized to -graphs, in which any two vertices are connected by a unique path of length . For no such graphs are known, and the claim of their non-existence is Kotzig's conjecture.
See also Central digraph, a directed graph with the property that every two vertices can be connected by a unique two-edge walk
References