Friendship graph

Friendship graph
The friendship graph F8.
Vertices	2n + 1
Edges	3n
Radius	1
Diameter	2
Girth	3
Chromatic number	3
Chromatic index	2n
Properties	Unit distance Planar Eulerian Factor-critical Locally linear
Notation	Fn
Table of graphs and parameters

The friendship graphs F₂, F₃ and F₄.

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

The friendship graph F_n can be constructed by joining n copies of the cycle graph C₃ with a common vertex, which becomes a universal vertex for the graph. ^[2]

By construction, the friendship graph F_n 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 graphs are generalized by the triangular cactus graphs.

Friendship theorem

The friendship theorem of PaulErdős , 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 colouring

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

${\displaystyle (x-2)^{n}(x-1)^{n}x}$.

The friendship graph F_n 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 theory

According to extremal graph theory, every graph with sufficiently many edges (relative to its number of vertices) must contain a ${\displaystyle k}$-fan as a subgraph. More specifically, this is true for an ${\displaystyle n}$-vertex graph if the number of edges is

${\displaystyle \left\lfloor {\frac {n^{2}}{4}}\right\rfloor +f(k),}$

where ${\displaystyle f(k)}$ is ${\displaystyle k^{2}-k}$ if ${\displaystyle k}$ is odd, and ${\displaystyle f(k)}$ is ${\displaystyle k^{2}-3k/2}$ if ${\displaystyle k}$ 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 ${\displaystyle k}$-fan. ^[10]

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 ${\displaystyle P_{k}}$-graphs, in which any two vertices are connected by a unique path of length ${\displaystyle k}$. For ${\displaystyle k\geq 3}$ no such graphs are known, and the claim of their non-existence is Kotzig's conjecture.

References

1. Weisstein, Eric W., "Dutch Windmill Graph", MathWorld
2. Gallian, Joseph A. (January 3, 2007), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics: DS6, doi: 10.37236/27 .
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 ${\displaystyle \scriptstyle 2^{\aleph _{\alpha }}}$ friendship graphs of cardinal ${\displaystyle \scriptstyle \aleph _{\alpha }}$", 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")", Metamath 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, vol. 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, vol. 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 .