7/23/2019 Friendship theorem

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THEOREM OF THE DAYThe Friendship TheoremIn a finite graph in which any two distinct vertices share exactly one commonneighbour some vertex is adjacent to all other vertices.

. . . in other words, if every pair of people at

a party shares exactly one mutual

friend at the party then some

guest is a friend of every-

body present. More isis true: friendships

must be arranged in

triangles which in-

tersect only in the

universally popular

guest. So the graphs

satisfying the hy-

pothesis of the theo-

rem are those shown

on the immediate

right, usuallyknown as

windmill

graphs.

Despite

the almost

trivial nature

of these graphs

the Friendship The-

orem seems surpris-

ingly resistent to dir-ect combinatorial proof;

and yet it emerges from a

few lines of linear algebra

as though it were the most

natural thing in the world.

Almost more interesting than the theorem

itself is the fact that it is false for infinite

graphs. Start withG0 = C5, the penta-

gon, as shown on the immediate left.

Now constructGn+1by creating

a new vertex adjacent to eachpair of vertices inGn having

no common neighbour. If

we repeat this indefin-

ately the result is

an infinite graph

satisfying the hy-

pothesis of the

theorem (for any

pair of vertices

we have, by defi-

nition, ensured a

common neigh-

bour) but not the

conclusion (since

it is clearly not a

windmill graph).

The graphs G0,

G1 and G2 are

shown on the

left;G 3 is shown

in the centre.A Challenge: try

to find a formula for the

number of vertices inGn: it is

100 forG3while forG4the number is 3695.

This theorem was proved by Paul Erdos,

Alfred Renyi and Vera Sos in 1966 inconnection with the problem of minimising

the edge set of a graph of fixed maximumdegree and smallest maximum distance.

Web link: the book proof is well described here: math.mit.edu/fox/MAT307-lecture20.pdf; for thefascinating full story see this MS Dissertation by Katie Leonard: web.pdx.edu/caughman/Katie.pdf.The Friendship Theorem appears as Theorem 6 near the end of the original paper: 1966-06 atwww.renyi.hu/p erdos/Erdos.html.

Further reading:Proofs from the Book, by Martin Aigner and Gunter M. Ziegler, Springer-Verlag, Berlin, 5th Edition, 2014, chapter 43.Created by Robin Whitty for www.theoremoftheday.org

http://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://www.renyi.hu/~p_erdos/1966-06.pdfhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.amazon.co.uk/gp/product/3662442043/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=3662442043&linkCode=as2&tag=robwhismatand-21http://www.theoremoftheday.org/http://www.theoremoftheday.org/http://www.theoremoftheday.org/http://www.theoremoftheday.org/Topology/Brouwer/TotDBrouwer.pdfhttp://www.theoremoftheday.org/Theorems.htmlhttp://www.theoremoftheday.org/CombinatorialTheory/Kuratowski/TotDKuratowski.pdfhttp://www.amazon.co.uk/gp/product/3662442043/ref=as_li_tl?ie=UTF8&camp=1634&creative=6738&creativeASIN=3662442043&linkCode=as2&tag=robwhismatand-21http://www.renyi.hu/~p_erdos/Erdos.htmlhttp://www.renyi.hu/~p_erdos/1966-06.pdfhttp://web.pdx.edu/~caughman/Katie.pdfhttp://math.mit.edu/~fox/MAT307-lecture20.pdfhttp://www.theoremoftheday.org/Resources/TheoremNotes.htm#163http://www.theoremoftheday.org/Resources/Glossary.htmhttp://www.theoremoftheday.org/