A number of graphs are associated with Geoffrey Exoo and Dan Ismailescu.
The graphs on 17, 19, and 21 vertices (Exoo and Ismailescu 2016, Soifer 2024) are examples of small triangle-free unit-distance graphs with chromatic number 4 and girth 4. These improve on larger examples previously found by O'Donnell, Chilakamarri, and Hochberg between 1994 and 1996, as summarized by Soifer (2008, Table 15.1, p. 445) and Soifer (2024, Table 15.1, p. 145) and reproduced below. Moreover, there is evidence that no smaller such graph exist (Soifer 2024). The 17-vertex Exoo-Ismailescu graph decorates the front cover of the October 2016 issue of Geombinatorics.
vertex count | graph | reference |
6448 | Wormald graph | Wormald (1979) |
56 | 56-O'Donnell graph | O'Donnell (1994) |
47 | Chilakamarri moth graph | Chilakamarri (1995) |
46 | 46-O'Donnell graph | unpublished (1995) |
40 | 50-O'Donnell graph | O'Donnell (1995) |
23 | Hochberg-O'Donnell fish graph | Hochberg and O'Donnell (1995) |
21 | 21-Exoo-Ismailescu graph | Exoo and Ismailescu (2016) |
21 | Kiteck-Payne graph | Kiteck and Payne (2021) |
19 | 19-Exoo-Ismailescu graph | Exoo and Ismailescu (2016) |
17 | 17-Exoo-Ismailescu graph | Exoo and Ismailescu (2016) |
Graphs on 40, 49, 51, 79, and 627 vertices are unit-distance graphs with girth 3 that appear in Exoo and Ismailescu (2018).
Graphs on 205 and 214 vertices appear in Exoo and Ismailescu (2019) as graphs labeled and , respectively.
Exoo-Ismailescu graphs are implemented in the Wolfram Language as GraphData["ExooIsmailescuGraph17"] etc.