[math-fun] Aknother knight's surprise?

Since you dispatched 4x4 so readily, perhaps you can manage something about the graph genus 6x6 knight moves on a torus instead? In the 4x4 case the vertices had valency 4 instead of 8 . The 6x6 case degenerates as well, less predictably: vertices opposite across the torus have all neighbours in common; identification yields 18 vertices and 36 edges and valency 4 immersed on a Klein bottle. The symmetry group has order 144 . Fred Lunnon