The friendship graph F8.
|Table of graphs and parameters|
The friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós (1966) 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 colouringEdit
Every friendship graph is factor-critical.
Extremal graph theoryEdit
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 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, in that for any smaller number of edges there exist graphs that do not contain a -fan.
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