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In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar undirected graph with 2n+1 vertices and 3n edges.[1]

Friendship graph
Friendship graph 8.svg
The friendship graph F8.
Chromatic number3
Chromatic index2n
Table of graphs and parameters
The friendship graphs F2, F3 and F4.

The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex.[2]

By construction, the friendship graph Fn is isomorphic to the windmill graph Wd(3, n). It is unit distance with girth 3, diameter 2 and radius 1. The graph F2 is isomorphic to the butterfly graph.

Friendship theoremEdit

The friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós (1966)[3] 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.[4]

A combinatorial proof of the friendship theorem was given by Mertzios and Unger.[5] Another proof was given by Craig Huneke.[6] A formalised proof in Metamath was reported by Alexander van der Vekens in October 2018 on the Metamath mailing list.[7]

Labeling and colouringEdit

The friendship graph has chromatic number 3 and chromatic index 2n. Its chromatic polynomial can be deduced from the chromatic polynomial of the cycle graph C3 and is equal to  .

The friendship graph Fn is edge-graceful if and only if n is odd. It is graceful if and only if n ≡ 0 (mod 4) or n ≡ 1 (mod 4).[8][9]

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.[10]


  1. ^ Weisstein, Eric W. "Dutch Windmill Graph". MathWorld.
  2. ^ Gallian, J. A. (January 3, 2007), "Dynamic Survey DS6: Graph Labeling" (PDF), Electronic Journal of Combinatorics, DS6, 1-58.
  3. ^ Erdős, Paul; Rényi, Alfréd; Sós, Vera T. (1966), "On a problem of graph theory" (PDF), Studia Sci. Math. Hungar., 1: 215–235.
  4. ^ Chvátal, Václav; Kotzig, Anton; Rosenberg, Ivo G.; Davies, Roy O. (1976), "There are   friendship graphs of cardinal  ", Canadian Mathematical Bulletin, 19 (4): 431–433, doi:10.4153/cmb-1976-064-1.
  5. ^ Mertzios, George; Walter Unger (2008), "The friendship problem on graphs" (PDF), Relations, Orders and Graphs: Interaction with Computer Science
  6. ^ Huneke, Craig (1 January 2002), "The Friendship Theorem", The American Mathematical Monthly, 109 (2): 192–194, doi:10.2307/2695332, JSTOR 2695332
  7. ^ van der Vekens, Alexander (11 October 2018). "Friendship Theorem (#83 of "100 theorem list")". (Mailing list).
  8. ^ Bermond, J.-C.; Brouwer, A. E.; Germa, A. (1978), "Systèmes de triplets et différences associées", Problèmes Combinatoires et Théorie des Graphes (Univ. Orsay, 1976), Colloq. Intern. du CNRS, 260, CNRS, Paris, pp. 35–38, MR 0539936.
  9. ^ Bermond, J.-C.; Kotzig, A.; Turgeon, J. (1978), "On a combinatorial problem of antennas in radioastronomy", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, pp. 135–149, MR 0519261.
  10. ^ Erdős, P.; Füredi, Z.; Gould, R. J.; Gunderson, D. S. (1995), "Extremal graphs for intersecting triangles", Journal of Combinatorial Theory, Series B, 64 (1): 89–100, doi:10.1006/jctb.1995.1026, MR 1328293.