The snail graph is the 29-vertex, 51-edge unit-distance graph obtained by adjoining two
vertices to the 27-vertex
configuration 
of Matolcsi et al. (2023), which satisfies 
. Both added vertices
have vertex degree 1 and are adjacent to the same
vertex of 
. Dúcz and Varga (2026) suggested the name from
the appearance of the resulting graph drawing.
The snail graph is a subgraph of an Exoo-Ismailescu graph and of several Heule graphs and Parts
graphs. Since 
is itself a subgraph of the snail graph, the same is
true of 
.
The snail graph has graph crossing number 5 and independence number 15. Its importance instead comes from its geometric fractional chromatic number. A geometric fractional coloring is a fractional coloring in which congruent subsets of the embedded vertices receive the same total weight of colors common to all their vertices. An exact rational dual certificate gives
 |
Thus 
itself has independence
ratio 
,
rather than less than 
.
Applying two finite blow-up procedures to its geometric fractional coloring data
produces a finite unit-distance graph with
independence ratio less than 
(Dúcz and Varga 2026). The order of the resulting
graph was not given and was described as astronomically large.
Dúcz and Varga (2026) report that ChatGPT and Codex assisted the software development and computational search, while the mathematical results and the verification program were developed and checked by the authors. Independent peer review had not been reported as of Sep. 21, 2026.
