The sailboat graph is the 16-vertex, 24-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph crossing number 2, graph diameter 4, and chromatic number 3. It is an apex graph and contains the drum graph as a topological minor.
The name appears to refer to the visual appearance of a sailboat in the embedding presented by Robertson et al. (2019, Fig. 14).
A cubic graph is cyclically 5-connected if it has girth at least 5 and every edge cut separating two subgraphs that contain cycles has at least five edges. It is dodecahedrally-connected if no 5-edge cut having at least seven vertices on each side cuts off a subgraph that can be drawn in a disk with its five attachment vertices on the boundary.
A graph is doubly-apex if identifying two of its vertices makes it planar. The characterization involving the sailboat graph states that a dodecahedrally-connected graph containing the sailboat graph but containing neither the Petersen graph nor any of the apex-selector graphs as a topological minor is doubly-apex (Robertson et al. 2019).
The sailboat graph is implemented in the Wolfram Language as GraphData["SailboatGraph"].
