The Goedgebeur graph is the 30-vertex cubic bipartite graph found by computer search by Goedgebeur (2015). It gives a counterexample
to the conjecture of Abreu et al. (2008) that the utility
graph 
,
Heawood graph, and Pappus
graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic
bipartite graphs. Here, a 2-factor is a spanning 2-regular subgraph, i.e., a disjoint
union of cycles covering all vertices. A graph is
pseudo 2-factor isomorphic if all its 2-factors have the same parity of number of
cycles.
The Goedgebeur graph has 45 edges, girth 6, graph diameter 5, cyclic edge connectivity
6, and automorphism group order 144. It has
312 2-factors, with cycle lengths 
, 
, 
, and 
(Goedgebeur 2015).
The name was introduced by Abreu et al. (2023).
The Goedgebeur graph is also the Levi graph of the self-dual 
Goedgebeur configuration.
Abreu et al. (2023) constructed the Goedgebeur graph from the Heawood graph and the Möbius-Kantor graph 
, the Levi
graphs of the Fano configuration and Möbius-Kantor configuration, respectively.
Aside from the Gray graph, the Goedgebeur graph is
the only known counterexample to the pseudo 2-factor isomorphic graph conjecture
(Abreu et al. 2025).
A number of minimal 8-crossing embeddings of the Goedgebeur graph are illustrated above (E. Weisstein, May 2-6, 2026).
The Goedgebeur graph admits two distinct order-1 LCF notation embeddings, both bilaterally symmetric, illustrated above (E. Weisstein, May 2, 2026).
Two curved-edge embeddings are illustrated above. The left drawing, due to Marién Abreu, displays the 
2-factor as the three ovals. The oval pattern suggested a possible infinite family
of analogous counterexamples, but the corresponding construction breaks down in larger
cases, underscoring the isolated nature of the Goedgebeur graph. The right drawing
is from Abreu et al. (2023).
Four unit-distance embeddings of the Goedgebeur graph are illustrated above (E. Weisstein, May 2, 2026).
The Goedgebeur graph will be implemented in a future version of the Wolfram Language as GraphData["GoedgebeurGraph"].
