Friendship graph explained

Friendship graph
Chromatic Number:	3
Girth:	3
Diameter:	2
Radius:	1
Properties:	Unit distance Planar Eulerian Factor-critical Locally linear

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.^[1]

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 ^[2] 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.^[3]

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

(x -2 ) ^{n} (x -1 ) ^{n} x .

The friendship graph is edge-graceful if and only if is odd. It is graceful if and only if or .^{[4] [5]}

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

k -fan as a subgraph. More specifically, this is true for an n -vertex graph if the number of edges is\left\lfloor (n ^{2})/(4) \right\rfloor + f(k), where f(k) is k ^{2-k} if k is odd, andf(k) is k ^{2-3k/2} if k 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 k -fan.^[6]

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

P _{k} -graphs, in which any two vertices are connected by a unique path of length k . For k\ge 3 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

Notes and References

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