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# Heawood graph

Percy John Heawood (1861-1955) was an English mathematician who
spent a large amount of time on questions related to the four colour
theorem.

There is a unique distance-regular graph Γ with intersection array
{3,2,2;1,1,3}. It has 14 vertices and spectrum
3^{1} (√2)^{6} (–√2)^{6} (–3)^{1}.
It is the point-line incidence graph of the Fano plane,
and is commonly called the Heawood graph.
It occurs as subgraph of the
Hoffman-Singleton graph.

It is the unique (3,6)-cage: the regular
cubic graph of girth 6 with minimal number of vertices.

### co-Heawood graph

The point-line nonincidence graph of the Fano plane is the
distance-3 graph of the above, and is distance-regular with
intersection array {4,3,2,1,2,4} and spectrum 4^{1}
(√2)^{6} (–√2)^{6} (–4)^{1}. It is sometimes called
the *co-Heawood graph*. It occurs as subgraph of the
Gewirtz graph, and is the
first subconstituent of the U_{3}(3) graph.

## Group

The full group of automorphisms is PGL(2,7) = L_{3}(2).2,
acting distance-transitively with point stabilizer S_{4}.

## Subgraphs

Substructures belonging to the maximal subgroups of the automorphism group:
a) *A partition of the edges into three matchings*.
There are 8 of these, forming a single orbit.
The stabilizer of one is 7:6,
with vertex orbit size 14.
(There are 24 matchings. The complement of a matching is a
14-cycle that decomposes uniquely into two matchings. So,
matchings come in groups of three.)

b) *A vertex*.
There are 14 of these, forming a single orbit.
The stabilizer of one is S_{4},
with vertex orbit sizes 1+3+6+4.

c) *An edge*.
There are 21 of these, forming a single orbit.
The stabilizer of one is D_{16},
with vertex orbit sizes 2+4+8.
These correspond to the flags of the Fano plane.

d) *A crossing non-edge*.
(The graph is bipartite; a crossing non-edge is a non-edge
meeting both sides of the bipartition.)
There are 28 of these, forming a single orbit.
The stabilizer of one is D_{12},
with vertex orbit sizes 2+6+6.
These correspond to the antiflags of the Fano plane.
The graph on the crossing non-edges, adjacent when
their union is a 4-coclique, is the
Coxeter graph.

## Map

Heawood proved the 7-color theorem for the torus.
This graph has an embedding on the torus with 7 areas that are
mutually adjacent, showing that 7 is best possible.

## References

P.J. Heawood,
*Map-colour theorem*,
Quart. J. Math. Oxford Ser. **24** (1890) 332-338.