The Georges graph, also called the Georges-Kelmans graph, is the 50-vertex nonhamiltonian cyclically 4-connected bicubic graph illustrated above. The graph was originally constructed by Kel'mans (1988), though rediscovered independently by Georges (1989) as the smallest member of an infinite family of nonhamiltonian 3-connected bicubic graphs. Gropp (1990, 1993) subsequently stated that the two graphs are isomorphic, and Kel'mans (1994) later gave a fuller account of his construction. Grünbaum (2009, p. 317) regarded the identification as unjustified from the published accounts, but the direct reconstruction given below confirms the isomorphism. Brinkmann et al. (2022) adopted the combined name Georges-Kelmans graph, credited Kel'mans and Georges with independent discovery, and proved that the graph is the smallest 3-connected bicubic nonhamiltonian graph.
The construction from two copies of a modification of the 18-Ellingham-Horton graph is illustrated above (Grünbaum 2006; Grünbaum 2009, pp. 310-311).
The Georges graph has girth 6 and is the Levi graph of the 
Georges configuration, for which Grünbaum
(2009, pp. 311-316) gave a geometric realization and Kocay (2010) subsequently
gave a realization with exact rational coordinates.
Kel'mans's two basic operations are shown above, reproducing his Figures 1 and 2. Given a cubic bipartite graph 
and two distinct edges 
and 
, the operation 
replaces each selected edge by a three-edge path
and attaches a pendant edge to each of the two new
internal vertices. The result is a four-pole 
whose four poles have degree 1. The second operation deletes
two specified edges 
and 
from the Petersen
graph and replaces them by four-poles 
and 
, attached as shown, to form the bicubic graph 
.
Let 
be the multigraph on two vertices with three parallel
edges 
,
, and 
, and let 
, with 
designated the middle edge. Form 
from two copies of 
whose middle edges are denoted 
and 
, and then set 
. Finally, form 
, where 
and 
are two copies of 
. This gives Kel'mans's 50-vertex graph 
.
The graph 
is illustrated above, with the two copies of 
shown in blue and green and the six remaining vertices of
the Petersen framework shown in orange. Direct computation shows that 
is isomorphic to the
Georges graph.
It is implemented in the Wolfram Language as GraphData["GeorgesGraph"].
with No Hamiltonian Circuit." Bull. Inst. Combin.
Appl. 46, 15-26, 2006.Grünbaum, B.