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Existence and uniqueness of the Hoffman-Singleton Graph (outline)
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Noam D. Elkies
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0. Let $G$\/ be a Moore graph of degree~$7$, i.e., a strongly
regular graph with parameters $(50,7,0,1)$, or equivalently
a graph of degree~$7$ on $50$ vertices with diameter~$2$ and girth~$5$.
Such $G$\/ contains $50\cdot7\cdot6\cdot6/10=1260$ pentagons.

1. Let $A$\/ be the neighborhood of one of those pentagons,
and let $B$\/ be its complement in $V(G)$.  Then $V(G)=A\cup B$\/
is a partition of $V(G)$ into two \hbox{$25$-vertex} parts
such that every vertex of~$G$\/ is adjacent to $2$ in the same part
and $5$ in the other part.  This is all we'll use about $A,B$.

2. Of the $1260$ pentagons, $1000$ are of type $ABxAB$ and 2$50$
of type $AAxBB$.  This leaves only $10$ of type $AAAAx$ and $BBBBx$.
It soon follows that each of $A,B$\/ consists of five pentagons.

3. We now know what the $AA$ and $BB$\/ edges look like;
the condition that $G$\/ have no 3- or \hbox{4-cycles} will force
the $AB$\/ edges uniquely up to automorphism of $A,B$.
First, each $A$ vertex has its one of its five $B$\/ neighbors 
in each of the five $B$\/ pentagons and vice versa.
Thus the restriction of $G$\/ to the union of
an $A$ and a $B$\/ pentagon is a Petersen graph.

4. We can orient the $A$ and $B$\/ pentagons compatibly,
i.e., label the $50$ vertices $A_i(x),B_j(y)$ ($i,j,x,y\in{\bf F}_5$)
so that the pentagon edges are $\{A_i(x),A_i(x+1)\}$ and
$\{B_j(y),B_j(y+1)\}$ and the $AB$\/ edges are
$\{A_i(x),B_j(2x+c_{ij})\}$ for some $c_{ij}$.  [A priori
it might have been necessary to use $-2x+c_{ij}$ instead
of $2x+c_{ij}$ for some $i,j$, but thanks to the girth condition
we can always flip some pentagons, i.e., relabel $A_i(x),B_j(x)$
as $A_i(-x),B_j(-x)$ for certain $i,j$, to eliminate all the
minus signs.]

5. To avoid 4-cycles it is now only necessary that
$c_{ij}+c_{i'j'}\neq c_{i'j}+c_{ij'}$ ($i\neq i',j\neq j'$).
This determines the $c_{ij}$ up to rotating the pentagons
[i.e., relabeling $A_i(x)$ as $A_i(x-\delta_i)$, which translates
the $c_{ij}$ by $\delta_i$, and likewise for $B_j(y)$] and permuting
the $i$'s and $j$'s among themselves.  For instance we may take
$c_{ij}=i\cdot j$.

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