Heawood Graph

The Heawood graph is a cubic graph on 14 vertices and 21 edges which is the unique (3,6)-cage graph. It is also a Moore graph. It has graph diameter 3, graph radius 3, and girth 6. It is cubic symmetric, nonplanar, Hamiltonian, and can be represented in LCF notation as . The Heawood graph is illustrated above in a number of embeddings.

The Heawood graph is isomorphic to the generalized hexagon , Knödel graph , and honeycomb toroidal graph . The line graph is the generalized hexagon .

It has chromatic number 2 and chromatic polynomial

Its graph spectrum is .

It is 4-transitive, but not 5-transitive (Harary 1994, p. 173).

The Heawood graph is one of eight cubic graphs on 14 nodes with smallest possible graph crossing number of 3 (another being the generalized Petersen graph ), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).

The Heawood graph corresponds to the seven-color torus map on 14 nodes illustrated above. The Heawood graph is the point/line Levi graph on the Fano plane (Royle).

Chvátal (1972) conjectured that point-line incidence graphs of finite projective planes, the smallest example of which is the Heawood graph, were not unit-distance embeddable. The first explicit embedding refuting this conjecture was found by Gerbracht (2008), and exactly 11 such embeddings (illustrated above) were published by Gerbracht (2009) following a general outline first suggested by Harris (2007).

An apparent unit-distance embedding based on a central hexagon has also been constructed by E. Gerbracht (pers. comm., Jan. 2010).

Another unit-distance embedding has apparently been found by Horvat (2009), illustrated above.

The Heawood graph is the second of four graphs depicted on the cover of Harary (1994).

It is implemented in the Wolfram Language as GraphData["HeawoodGraph"].