The log graph is the 20-vertex, 30-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph crossing number 2, graph diameter 4, and chromatic number 3. It is not an apex graph.
The log graph was introduced as one of nine forbidden topological minors characterizing the apex graphs within a highly connected class of cubic graphs (Robertson et al. 2019).
The log graph is implemented in the Wolfram Language as GraphData["LogGraph"].
